The very short answer is that the "elect" are the saints, the Christians.
But, this question encompasses the destruction of Jerusalem, and the prophesies from Daniel and Revelation. We have to identify the context, and the time in order to be able to identify the elements of the prophetic language in Matt. 24:29-31.
In Matt. 24:2, Jesus had just told the disciples that the beautiful temple was going to be torn down. The main building of the temple had been completed before Herod died, but the rest of the buildings were still undergoing reconstruction while the disciples spoke to Jesus.
So, the news that the temple was to be torn down was an astounding statement to the disciples, indeed to any Jew, as not only were these massive buildings, but the temple was the center of their lives. If it was going to be torn down, then in their minds it meant the end of their world as they knew it. (See more information at Herod's Temple)
Understandably, they asked Christ in verse 3 when that temple would be torn down, and what would be the signs. They recognized enough to attribute the destruction of the temple as a judgment from God, and continued to ask about the signs of His coming. Think "day of the Lord" - and of the end of the "world", and a time of judgment.
"And as he sat upon the mount of Olives, the disciples came unto him privately, saying, Tell us, when shall these things be? and what shall be the sign of thy coming, and of the end of the world?" (KJV)
The correct word in verse 3 that is translated as "world" in the KJV is 'αἰῶνος', or "aionos", and in the English is "age". An age is a cycle of time. Time is an element of the "age."
Excerpt from Benson Commentary on Matt. 24:3-5,
"These seem to be only different expressions to denote the same period with the destruction of Jerusalem, the disciples supposing, that when the destruction of Jerusalem should take place, then would be the coming of Christ and the end of the world, or, the conclusion of the age, as συντελεια του αιωνος should rather be translated here, and often signifies. See especially Hebrews 9:36; and 1 Corinthians 10:11. Accordingly, in the parallel place of Mark 13:4, their question is expressed thus, When shall these things be, and what shall be the sign when all these things shall be fulfilled? And in that of Luke 21:7, thus, When shall these things be? and what sign will there be when these things shall come to pass? The disciples therefore inquire concerning two things; 1st, the time of the destruction of Jerusalem; and, 2d, the signs of it. The latter part of the question our Lord answers first, and treats of the signs of his coming from the 4th to the 31st verse, inclusive; and then passes on to the other part of the question, concerning the time of his coming." Source: here
We know that the temple was destroyed in A.D. 70 in the Jewish-Roman wars. So, the end which Jesus was telling the disciples to watch for was going to be during that time period. The subject of the question was the destruction of that temple. Therefore, the end they were to watch for, and the signs they were to watch for was for the end of the temple, and the end of the old sacrificial covenant.
Matt. 24:34 confirms the time period of this "end" was their generation.
"Verily I say unto you, This generation shall not pass, till all these things be fulfilled." (KJV)
"This generation" was their generation, the generation in which Christ spoke the words. Not the generation in which we are reading the book; but, the generation in which Christ was manifested on this earth in the first century A.D.
Therefore, the prophesy language of Matt.24:29-31 is of that time period for the destruction of the temple and of Jerusalem in the days leading up to the wars in Judea AD 66 to AD 70. This is important because it is a time marker for the prophesies Christ references in these verses.
Verse 29 references Daniel's prophesy of the tribulation.
" Immediately after the tribulation of those days shall the sun be darkened, and the moon shall not give her light, and the stars shall fall from heaven, and the powers of the heavens shall be shaken:" (KJV)
" And at that time shall Michael stand up, the great prince which standeth for the children of thy people: and there shall be a time of trouble, such as never was since there was a nation even to that same time: and at that time thy people shall be delivered, every one that shall be found written in the book." (KJV)
In the OT, Michael prefigured Christ as the warrior for His people, and is so referenced again in Rev. 12:7. The prophesy to Daniel was about Daniel's people - the Jews. So the deliverance of Dan. 12:1 from the trouble or tribulation was referenced by Christ in Matt. 24:29 in the prophetic language of rulers being removed from their stations of power.
That Christ referenced Daniel in this prophesy of the destruction of the temple marks Daniel's prophesy for that same generation of the 1st century AD.
Christ also told them the "end" would be "immediately" after the tribulation of those days. In other words, there would be no delay between the tribulation - the persecution and the war - and the destruction of the temple.
Excerpt from Benson Commentary:
" Immediately after, &c. — We are now come to the last act of this dismal tragedy, the destruction of Jerusalem, and the final dissolution of the Jewish polity in church and state, which our Lord, for several reasons, might not think fit to declare nakedly and plainly, and therefore chose to clothe his discourse in figurative language. Commentators, indeed, have generally understood this, and what follows, of the end of the world, and of Christ’s coming to judgment: but the words, immediately after the tribulation of those days, show evidently that he is not speaking of any distant event, but of something immediately consequent upon the tribulation before mentioned, and that must be the destruction of the temple and city of Jerusalem, and the abolition of the Jewish polity, civil and religious." Source: here
Darkening of suns and moons was God's standard prophetic language throughout the OT for the downfall of kings and queens, princes or whatever ruling authority governed the people.
The people under discussion in Matt. c 24 were the Jews, Daniel's people (Dan. 9:24). The same people to which Christ appeared, and to whom He was speaking. The stars falling from heaven were the tribes of Israel, which are defined throughout the OT prophetic language.
God promised Abraham in Gen. 22:17,
" that blessing I bless thee, and multiplying I multiply thy seed as stars of the heavens, and as sand which [is] on the sea-shore; and thy seed doth possess the gate of his enemies;" (YLT)
While speaking to Isaac, God repeated the promise He gave to Abraham in Gen. 26:4,
"and I have multiplied thy seed as stars of the heavens, and I have given to thy seed all these lands; and blessed themselves in thy seed have all nations of the earth;" (YLT)
Moses reminded God of His promise in Ex. 32:13,
" `Be mindful of Abraham, of Isaac, and of Israel, Thy servants, to whom Thou hast sworn by Thyself, and unto whom Thou speakest: I multiply your seed as stars of the heavens, and all this land, as I have said, I give to your seed, and they have inherited to the age;'" (YLT)
The "stars of heaven" were the children of Abraham, Isaac, and Jacob / Israel. Joseph's dream in Gen. 37: 9-10 identify his brothers - the other children of Israel - as stars.
So, the falling stars in the prophesy of Matt 24, just as in many others from the OT were the children of Abraham, Isaac, and Israel.... the tribes of Israel, and specifically those of the priesthood and the Sanhedrin.
Prophetic language that speaks of the "powers of heavens being shaken" does not speak of the heaven where God sits on His throne. That heaven cannot be shaken. The "heavens" that can be shaken are the earthly kingdoms of men, who rule over the people. They are a type of heaven, as they have authority and station above the common man.
Just as God from His heavenly throne has authority over all men, the earthly kings rule from their "heavens" of their kingdoms on earth. We have to be careful to identify which "heavens" are being spoken of in prophesy. If the "heavens" can be shaken, then they are the ruling authority or kingdoms of men.
Therefore, shaking the "powers of heavens" in Matt. 24:29 meant the removal or downfall of the ruling authorities of the Jews, specifically the Sanhedrin and the priesthood who tyrannically ruled the lives of those in Jerusalem and Judea.
It may also reference the next "heavens" or authority over them, as the Sanhedrin answered to Rome. At that time, Judea was a province of Rome. Rome was going to come under judgment as well during the Year of Four Caesars in A.D. 69 - 70, and afterwards.
See similar language in Ezek. 32:7-8; Joel 2:30-31.
Matt. 24:30 spoke of Christ's coming in the prophetic judgment language against Jerusalem and that old temple in the same manner in which Christ had told Caiaphas that he would see the Son of Man "coming upon the clouds of the heaven, with power and much glory;"
" Jesus saith to him, `Thou hast said; nevertheless I say to you, hereafter ye shall see the Son of Man sitting on the right hand of the power, and coming upon the clouds of the heaven.'" (YLT)
This is the same prophetic language of Dan. 7:13 of the ascension of Christ,
" I saw in the night visions, and, behold, one like the Son of man came with the clouds of heaven, and came to the Ancient of days, and they brought him near before him." (KJV)
Coming in the clouds was a time of judgment upon the people and nation(s). See also Luke 21:27.
The sound of the trumpet in In Matt. 24:31 was the announcement of His coming and the gospel call. The angels are messengers, and could be those heavenly beings that carry out God's instructions, but the word is also used for any agent of God, including the apostles who were leading the people of God during this time, preaching the gospel throughout all the Roman empire (rule or "heavens").
Excerpt from Gill's Exposition:
" With a great sound of a trumpet, meaning the Gospel; see Isaiah 27:13 so called in allusion either to the silver trumpets which Moses was ordered to make of one piece, and use them for the calling of the assembly, the journeying of the camps, blowing an alarm for war, and on their solemn and festival days, Numbers 10:1. The Gospel being rich and precious, all of a piece, useful for gathering souls to Christ, and to his churches;...." Source: here
The gathering of His elect was the gathering of His saints into a safe place, away from the judgment / battle on the "earth" of Judea. It is seen in Daniel's prophesy.
Dan. 7:26-27, speaking of the fourth kingdom in which Christ would be manifested,
" But the judgment shall sit, and they shall take away his dominion, to consume and to destroy it unto the end. 27 And the kingdom and dominion, and the greatness of the kingdom under the whole heaven, shall be given to the people of the saints of the most High, whose kingdom is an everlasting kingdom, and all dominions shall serve and obey him." (KJV)
The coming of the Lord in Matt. 24 was the same as "judgment shall sit" in Dan. 7:26.
The elect are His saints, those who believe in Him, who have chosen to obey the command to be immersed in His name - baptized in His name, and are then written in the book of life (Phil. 4:3; Rev. 3:5; Rev. 21:27).
The election is made on both sides of the agreement: the first of the offer from God, the call from God to obey the gospel of Christ; and the second on behalf of the ones who answer that call. It is a free will gift from God (grace), and a free will acceptance by each who will obey the command. Therefore, "the elect".
Those "elect" or saints of the first century AD that were going to have to go through that tribulation would be safeguarded by the angels / messengers / apostles and be removed from the danger of the Jewish-Roman wars. Christ had told them in Matt. 24:16 to flee to the mountains when they saw certain signs. See also Luke 21:21-22.
(Note: It does no good to flee from a world-wide destruction such as Noah's flood, so this chapter and Christ's warnings were not speaking of the end of all the physical cosmos / world.)
The elect who heeded the warnings are the ones who would be "taken" out of danger in verses Matt. 24:40-41. Those who heeded the warnings fled when they saw the signs Christ spoke of: the Roman armies, the false prophets, etc.
The ones who did not heed the warnings - those left behind - faced the judgment battle, and suffered injury, enslavement, or death.
Those taken were not some rapturous removal in to the sky. They were the ones who heeded Christ's warning to flee to the mountains. Tradition has it that they all safely fled to Pella, or the surrounding area.
Eusebius recorded that
"The members of the Jerusalem church by means of an oracle, given by revelation to acceptable persons there, were ordered to leave the city before the war began and settle in a town in Peraea called Pella." Book III, 5:4.
Epiphanius (375 AD):
"The Nazoraean sect exists in Beroea near Coele Syria, in the Decapolis near the region of Pella, and in Bashan in the place called Cocaba, which in Hebrew is called Chochabe. That is where the sect began, when all the disciples were living in Pella after they moved from Jerusalem, since Christ told them to leave Jerusalem and withdraw because it was about to be besieged. For this reason they settled in Peraea and there, as I said, they lived. This is where the Nazoraean sect began." (Panarion 29:7:7-8)
Eutychius of Alexandria (10th c AD)
"Qistus, governor of Jerusalem, died and the city was without any authority or sovereign to govern it. The Jews then arose and rioted and killed James, son of Joseph, known as the “brother of the Lord”, stoning him to death (14). Then they harassed a group of disciples and expelled them from the city. The Christians abandoned Jerusalem, crossed the Jordan and settled in those places (15). Informed of this fact, Nero Caesar sent word to the commander stationed in the East, named Vespasian, to rally his troops and go to Judea with orders to kill all the inhabitants, sparing none, and to destroy the houses." The Annals of Eutychius of Alexandria, ch. 9:
John Lightfoot (1889):
"Jerusalem was taken in the autumn of 70 A.D. Before its fall the Christians had left the doomed city. While the greater part retired beyond the Jordan and founded Christian colonies at Pella and the neighborhood, the principle leaders of the church -- the surviving apostles and other personal disciples of the Lord -- sought a new home in proconsular Asia. Henceforward we find the headquarters of Christendom no more at Jerusalem, nor even at Antioch but, (for the time at least) in Ephesus. Here John fixed his abode after his temporary banishment in Patmos." (Lightfoot, J.B.; Translated and edited; Apostolic Fathers: Clement, Ignatius, Polycarp; first published 1889; Pub. Hendrickson; Vol. 1, pg. 438.)
This was the same flight that Christ prophesied to John in Rev. 12:6 of the "woman" (the church of Christ) who fled the "dragon" (Rome) into the wilderness for 1260 days, or 3-1/2 years. The revolt began in Jerusalem about Oct. 66 AD, and after Cestius Gallus failed to put down the rebellion here, Nero sent Vespasian who began his campaign in early 67 AD marching inward from Galilee and working his way through Judea here.
In AD 69 Vespasian left his son Titus in charge of the campaign, and gathered his supporters to return to Rome and take control of the falling empire during the year of 4 Caesars, as the destruction in Rome under Galba, Otho, and Vitellius had almost destroyed the city.
Meanwhile, Titus had reached Jerusalem in March / April AD 70, waited until the gates of the city were shut for the Passover, and laid siege. Six months later, approx. Sept. 8-10 AD 70 Jerusalem fell, and the temple was destroyed.
" The siege of Jerusalem was probably the greatest single slaughter in ancient history." J.E. Lendon, Military History Quarterly Summer 2005, The Roman Siege of Jerusalem here
Those that had fled Jerusalem and the battle areas in Judea when they saw the Roman armies approaching sheltered in Pella and other areas beyond the Jordan. Many then moved on and settled in different areas after the destruction was over.
So, after A.D. 70, the scattered Jewish Christians all dispersed throughout the nations, which furthered the teaching of the gospel of Christ to all the world.
" ...for out of Zion shall go forth the law, and the word of the Lord from Jerusalem." (KJV)
" The Lord also shall roar out of Zion, and utter his voice from Jerusalem; and the heavens and the earth shall shake: but the Lord will be the hope of his people, and the strength of the children of Israel." (KJV)
Christ, our Lord roared from Zion (Jerusalem) during His earthly ministry, at His death on the cross, when He was resurrected and appeared to the disciples, when He ascended to heaven, and when He sent them the Comforter, and poured out the Holy Spirit upon the people of the first century A.D. The Holy Spirit's record of His testimony is still roaring today!
The judgment of Jerusalem, and the tribes of Israel was fulfilled in the destruction of that temple in A.D. 70, and the scattering of the once chosen people of God. See Ezek. 22:20-22; Dan. 12:7.
Today, the chosen are the elect - the saints - who have put on Christ through baptism (immersion). When we are baptized into Christ (Mark 16:16), we are covered by His blood (Ex. 12:13; Rom. 5:9), and are clothed in His righteousness (the wedding garments of Matt. c. 22). Then we are counted for the seed of Abraham (Gal. 3:26-29), and are children of faith (Rom 4:10-18), and we become the Israel under the new covenant of the gospel of Christ which is saved with an everlasting salvation (Isa. 45:17).
Those of us who are in Christ are Israel.
Through the gospel of Christ, our heavenly Father has given us everything we need to attain salvation (Acts 4:12; 2 Pet. 1:3-4). We are not waiting for Him to do anything more. We are assured through His word that He has already accomplished the predetermined plan of salvation (1 Pet. 1:20; Matt. 25:34; Rev. 13:8) in His sacrifice on the cross, and His fulfillment of the old covenant, and His destruction of that old sacrificial temple.
We rejoice greatly because He did it all! He came back to those of the first century A.D. just as He promised them that He would. He continues to sit in judgment, and rules from the right hand of the Father over all of His kingdom. He continues to judge the nations from heaven above as He has always done. He doesn't have to come back to earth to do His will, or to execute any necessary judgment.
Today, we live our lives according to His will and remain faithful unto death. Then, we will hear our last trump. We will be resurrected, changed in the twinkling of an eye (1 Cor. 15:51-54), and be gathered together into heaven with all of the saints that have gone before us. 1 Thess. 4:16-17:
" 16 For the Lord himself shall descend from heaven with a shout, with the voice of the archangel, and with the trump of God: and the dead in Christ shall rise first:
17 Then we which are alive and remain shall be caught up together with them in the clouds, to meet the Lord in the air: and so shall we ever be with the Lord."
The tendency of most people when reading the above two verses is to link them in time. The word "then" in verse 17 makes a distinction in time for those that remained alive... those who had not yet fallen alseep... when the Lord came back to them in A.D. 70.
"Then,"... afterwards ...after the judgment against Israel, and after the destruction of the temple. It indicates that afterwards life was going to continue on; that after the Lord's second appearance in that generation (Heb. 9:28) the living would be taken up as they died.
After the temple was destroyed, Christ opened the gates of Hades (Rev. 1:18) and carried out the first resurrection of all of the dead who had been waiting in Hades (Matt. 25:31-33; Rev. 20:12), all those that had died before His judgment coming in A.D. 70. (See the my post at ShreddingTheVeil - "The Burning of Jerusalem and the Hadean Death.")
Then (after), the resurrection process changed from a group or general resurrection to the individual resurrection.
"And I heard a voice from heaven saying unto me, Write, Blessed are the dead which die in the Lord from henceforth: Yea, saith the Spirit, that they may rest from their labours; and their works do follow them." (KJV)
Today, Christ is still sitting on His throne. His judgment has never stopped; it is on-going. For all that remained alive at His second appearance, all that had not yet died would be caught up as they passed from mortal to immortal life to be gathered into heaven to be with the Lord forever more.
"Caught up together" means being taken to where they are in heaven, to be with them. It does not mean at the same time. Any version of the translation that reads "simultaneously" is in error.
Ever since the judgment of A.D. 70, the gathering of the elect has been happening every hour of every day that a soul in Christ passes from this corruptible, mortal life into everlasting, eternal life in heaven.
There are many sources that can be reviewed for the history of the Jewish wars with Rome. Here are a few: Causes of the War; Titus' Siege of Jerusalem;and Josephus' War of the Jews, Book 4
All bold emphasis is mine.
There is much more scriptural evidences of Christ's soon return in that same generation of the first century AD for the judgment of Jerusalem and the destruction of the temple at my blog ShreddingTheVeil. All ten parts of "It's Not The End of The World", the eight parts of "The Signs of Revelation", "Daniel and The End Times", "The Whore of Babylon," "God's Definitions Part I - The House of Israel" and others expand on this theme.