Some good questions to answer before exegeting the passage you've quoted include the following:
What is the time frame indicated? Is it now or at some time in the future when this reigning will take place?
Who are the "we" in the passage?
Is the "reigning" in this passage related to Jesus' announcement of the "Kingdom of God" (or the "Kingdom of Heaven")?
There are perhaps other questions we could ask which would aid us in answering what appears to be your central question (viz., what does reigning involve? and who are the subjects of those who reign?), but these three seem to me to be foundational.
Now, or Then?
I opt for then. Reigning is part of kingdom vocabulary, to be sure. Jesus announced a kingdom, but that kingdom resided within each believer:
"Once, having been asked by the Pharisees when the kingdom of God would come, Jesus replied, 'The kingdom of God does not come with your careful observation, nor will people say, "Here it is," or "There it is," because the kingdom of God is within you'" (Luke 17:21-22 NIV).
In other words, Jesus seems to be teaching that the reign of God's kingdom is within the heart of each believer. Each believer is a subject of the King. Jesus does not seem to be teaching that the King of the Kingdom of God is like any earthly potentate. His kingdom is not maintained through military might and the threat of force.
"Jesus said, 'My kingdom is not of this world. If it were, my servants would fight to prevent my arrest by the Jews. But now my kingdom is from another place'" (John 18:36 NIV).
The phrase "from another place" seems to indicate an other-worldly kingdom, and its reign which is currently in the hearts of its subjects will undergo a transformation when it appears in its fullness to the entire world, and not just to the King's believers. John's Revelation of Jesus Christ speaks of that aspect of the Kingdom of God:
"And the seventh angel sounded; and there were great voices in heaven, saying, 'The kingdoms of this world are become the kingdoms of our Lord, and of his Christ; and he shall reign for ever and ever'" (11:15 KJV).
Who are "We"?
"We" are believers who "endure." Endure what? Endure the hardships which come from being a subject of the King now, in this life, not the next. We find the theme of endurance in many places in the New Testament, and biblically speaking, endurance (or perseverance, 2 Peter 1:6 and Romans 5:4) is the product of a training process. Much as an athlete develops endurance through rigorous physical training, so do believers develop their spiritual muscles in Jesus' boot camp. Spiritually, that training process for believers includes trials, tribulations, and dying to self, or as Jesus put it,
"'If anyone wants to come with Me, he must deny himself, take up his cross, and follow Me'" (Matthew 16:34; see also Matthew 10:38; Mark 8:24 and 10:21; and Luke 9:23).
Jesus did not promise a primrose path to his disciples. He did, however, promise rewards to those who 1) endure patiently everything the world throws at them for being his followers; and 2) serve others faithfully, just as Jesus came not to be served, but to serve and to give his life as a ransom for many (see Matthew 20:28 and Mark 10:45).
The Reign of God's Kingdom
The reign of God's Kingdom during the church age, during which Christ is building his kingdom one soul at a time until his work of building we be completed, will experience a transformation. At that time, the exact time of which is shrouded in mystery, the Kingdom's subjects will then become co-regents with the King of kings and Lord of lords.
The Greek word for reign is the word from which we derive our word basilica. A basilica in ancient Rome was a public building which was used as a courtroom, an assembly hall, and an administration building for conducting the business of the empire. (Related to the word reign is another Greek word, basileus, or king.)
At some future time, all those who have been disloyal to the King in this life will have been judged at the Great White Throne (see Revelation 20:11 ff.). If at this judgment their names are not found written in "the book of life," they will be thrown into the lake of fire, also known as hell.
Since each believer will also be judged at the judgment seat of Christ (the bema; see Romans 14:10-12 and 2 Corinthians 5:10) and be rewarded by Christ for the good works performed during his or her lifetime, we can safely assume that each person's reward will not be the same.
Since at the judgment seat of Christ, believers' works will be tested by fire, only those works which Paul refers to as gold, silver, and precious stones (1 Corinthians 3:12-15) will survive the purifying process. For this reason, the true motives for those "good works" will be exposed. If the motives originated in self-aggrandizement, pride, selfishness, or some other base desire, those works will not survive the fire of judgment.
On the other hand, those works which emerge from the fire unscathed will be those which were performed out of love for the Savior and for people. These are the gold, silver, and precious stones.
All this to say, since the Kingdom of God will be populated by saints from every tongue, tribe, and nation, with the exact number of saints unknown but which will one day comprise a great throng (see Revelation 7:9-10), each saint will be given work to do within the God's eternal kingdom.
The "basilica," as it were, for this kingdom in whatever form it takes, will be the scene of the administration of God's kingdom. I suggest that those who endured a great deal as believers in the kingdom of the heart will have more responsibilities than those who endured little. Likewise, those believers whose good works survive the fire of judgment at the bema, will also be entrusted with much, whereas those believers, whose works are mostly burned up in the fire of judgment, will be entrusted with little.
Jesus' teaching on the theme of "to whom much is given is much required" (see Luke 12:48) as well as his parable of the talents (see Matthew 25:14 ff.) are of some relevance in this regard. Accordingly, God expects a return on his investment in each of his children. Since we do not all have the same gifts, he does not expect as big a return from those possessing modest gifts as he does from those possessing greater gifts. He does, however, expect faithfulness from each person, regardless of gifts, in using them for his glory.
I suggest that believers look forward with great anticipation to hear those words from the Master,
"Well done, good and faithful slave. You were faithful with a few things, [so] I will put you in charge of many things; enter into the joy of your master [or lord]" (Matthew 25:21 and 23).
The key words in these verses vis a vis your question are in charge of. A person in charge of many things is a person who reigns over, or administers, the affairs of the Master. Through his teaching in the parable of the talents Jesus seems to be saying that during the eternal reign of God in the new heaven and the new earth (Revelation 21:1), those who were once subjects of the King will then be both subjects and co-regents of the king.
In conclusion, all believers in the eternal kingdom, as it will be established in the new and restored earth, will be subjects of the King. God's enemies will have been put under his feet, and humankind's last enemy, death, will have been abolished. Then the Father and his Christ, along with those saints who endured to the end, will reign eternally to God's glory. Here is how Paul envisions God's reign on the new earth:
"And when all things have been subjected unto him [i.e., the God and Father], then shall the Son also himself be subjected to him that did subject all things unto him, that God may be all in all. For, He put all things in subjection under his feet. But when he saith, All things are put in subjection, it is evident that he is excepted who did subject all things unto him. And when all things have been subjected unto him, then shall the Son also himself be subjected to him that did subject all things unto him, that God may be all in all" (1 Corinthians 15:24-28 NAS).
In conclusion, I envision God's eternal kingdom embodying a perfect egalitarianism. Since all the King's subjects owe their existence in that kingdom to the grace of God, no one who is there deserves to be there. Along with this perfect egalitarianism, however, will be a heavenly hierarchy of redeemed saints, if you will, through whom God will reign (there is that word again!) and administer the affairs of his kingdom. Christ's parable of the talents hints at this when the master in the story says,
"I will put you in charge of many things" (op. cit.).
Within this heavenly hierarchy, unlike every earthly hierarchy, there will be no jockeying for position, no resentment of one's "superiors," and no "passing of the buck" in doing God's will. Each of us will be occupied with worshiping the King by accomplishing the King's bidding with joyful hearts. The work we do will bring fulfillment to us and glory to God as together we travel the universe, possibly, discovering new and thrilling aspects of God, of His creation, and of His kingdom.
Then and only then will God's will be done perfectly "on earth as it is in heaven" (Matthew 6:10).