The Idea in Brief
The "Times of the Gentiles" is the period of time when the visible theocratic kingdom on earth is absent. That is, when the visible theocratic kingdom on earth existed, the Biblical referent of time was in reference to what Jewish ruler(s) were in power. When the theocratic kingdom on earth ended with the departure of the glory of the Lord (before the Babylonian Captivity), the Hebrew Bible then referenced Gentile powers as the referent of time. This Gentile rulership began at the time of the Babylonian Captivity, and continues to this day, and therefore the fulfillment of the Times of the Gentiles awaits a future day, when the visible theocratic kingdom will be established again on earth.
In the Hebrew Bible, the theocratic kingdom began at Sinai, when the glory of Yahveh dwelt (glory-dwelled = "shekinah") in the Tabernacle and later in Solomon's Temple. First Moses, the judges, and then the Jewish kings were the representatives of the leadership of this theocratic kingdom on earth. That is, Yahveh was the Head of State.
Therefore the referents of time in the Hebrew Bible were not based on the age of the world, but based on the reigns of these respective Jewish leaders at the time. For example, Hebrew Scripture references the reigns of the Jewish kings as time referents, since they had represented the leadership of the theocratic kingdom (even when divided).
1 Kings 16:23 (NASB)
23 In the thirty-first year of Asa king of Judah, Omri became king over Israel and reigned twelve years; he reigned six years at Tirzah.
That is, the rule of Jewish kings was the basis of gauging time, since their rulership was the nominal extension of the authority of Yahveh, who was the Head of State of the theocratic kingdom, even if divided. The laws of this kingdom came from the Torah.
However, when the Babylonian Captivity occurred, the glory of Yahveh had departed the Temple and this theocratic kingdom on earth ended: thus the reign of Jewish kings had terminated (and so there was no longer any more referent of time based on the reign of any Jewish kings). At this time the prophet Daniel introduced the images of reigning Gentile powers on earth as rapacious animals, which one day would end. So post-exilic Hebrew Scripture such as Zechariah and Haggai reference the reigns of these Gentile powers (and not Jewish kings), and so these references are to the times of the reference of reigning Gentile powers on earth versus any Jewish power -- thus "the Times of the Gentiles." The following graph provides an overview of "the Times of the Gentiles."
The last eight chapters of Ezekiel (approximately 20% of the book) concerns dozens of measurements, which suggest the literal establishment of the new temple, where the glory of Yahveh will one day dwell. (The dozens of references to precise measurements with a ruling measuring rod suggest that this temple is literal.) It was in this context of the apocalyptic end that Jesus was alluding in Luke 21:24 to the fulfillment or completion of the "Times of the Gentiles," when Gentile rule would end in the world in general and over the Promised Land in particular. That is, Jesus was alluding to the time when these rapacious Gentile powers would no longer dominate the earth.
At the current time (today) we are living in the invisible kingdom of God (under the New Covenant), where the temple is the body of the believer in the individual and collective sense. That is, the believer operates in the kingdom of God while on earth (please see Acts 26:16-18 and Col 1:13). Since this kingdom is invisible, there is no exercise of direct theocratic authority on earth. That is, there is no direct exercise of the Torah within this invisible kingdom.
So the "Times of the Gentiles" refers to this indefinite period of time on earth, which will one day end when the Lord comes back to earth to reestablish his rule and authority on the earth, when "the nations will be broken with a rod of iron and shattered like earthenware" (Ps 2:9 with Rev 2:26-27, Rev 12:5, and Rev 19:15). The Apostle Paul is making the same reference to the fulfillment of the Times of the Gentiles, which will inaugurate the Zionic era.
Rom 11:25-27 (NASB)
25 For I do not want you, brethren, to be uninformed of this mystery—so that you will not be wise in your own estimation—that a partial hardening has happened to Israel until the fullness of the Gentiles has come in; 26 and so all Israel will be saved; just as it is written,
“The Deliverer will come from Zion,
He will remove ungodliness from Jacob.”
27 “This is My covenant with them,
When I take away their sins.”
In conclusion, Scripture points to the literal establishment of the theocratic kingdom on earth at some point in the future, which will be centered on the Temple described by Ezekiel (and which translates into the eternal Zionic Jerusalem described in Revelation 21:10-27, where the measuring rod appears again suggesting literal events and objects).
The "Times of the Gentiles" is the period of time when the visible theocratic kingdom on earth is absent.