Q:

What is meant by the "fullness of the Gentiles" or the "times of the Gentiles"?

In Luke 21:24, Jesus speaks of future events, including the destruction of Jerusalem and His return, saying:

“Jerusalem will be trampled underfoot by the Gentiles, until the times of the Gentiles are fulfilled” (ESV).

A similar phrase is found in Romans 11:25, which says, 

A partial hardening has come upon Israel, until the fullness of the Gentiles has come in (ESV).

What did Jesus and Paul mean by the “times of the Gentiles” or the "fullness of Gentiles"?

A:

Does the Bible tell us what the phrase “times of the Gentiles” or "fullness of Gentiles" means?
Often the Bible says something that offers a glimpse of something that raises many more questions than it answers.

Luke 21:20-24  And when ye shall see Jerusalem compassed with armies,
  then know that the desolation thereof is nigh.   Then let them which
  are in Judaea flee to the mountains; and let them which are in the
  midst of it depart out; and let not them that are in the countries
  enter thereinto.   For these be the days of vengeance, that all things
  which are written may be fulfilled.   But woe unto them that are with
  child, and to them that give suck, in those days! for there shall be
  great distress in the land, and wrath upon this people.   And they
  shall fall by the edge of the sword, and shall be led away captive
  into all nations: and Jerusalem shall be trodden down of the Gentiles,
  until the times of the Gentiles be fulfilled.

The context of Luke 21:24 is a description Jesus is giving about the "days of vengence". There are some other verses that bear upon this time.

Daniel 9:26-27  And after threescore and two weeks shall Messiah be
  cut off, but not for himself: and the people of the prince that shall
  come shall destroy the city and the sanctuary; and the end thereof
  shall be with a flood, and unto the end of the war desolations are
  determined.   And he shall confirm the covenant with many for one
  week: and in the midst of the week he shall cause the sacrifice and
  the oblation to cease, and for the overspreading of abominations he
  shall make it desolate, even until the consummation, and that
  determined shall be poured upon the desolate.

That this can be taken to describe what Jesus is talking about can be seen from the parallel verses in Matthew;

Matthew 24:14-16  And this gospel of the kingdom shall be preached in
  all the world for a witness unto all nations; and then shall the end
  come.   When ye therefore shall see the abomination of desolation,
  spoken of by Daniel the prophet, stand in the holy place, (whoso
  readeth, let him understand:)   Then let them which be in Judaea flee
  into the mountains:

From Daniel we get an understanding that after the arrival of Jesus there will be a seven year (a "week" being shâbûa "sevened" in Hebrew) interruption by the "prince" that shall come.
Some feel that everything described in Daniel, Matthew, Luke , and Revelation all happened at the destruction of the temple in 70 AD. I think you have to reach too far to advance that theory.
I think an allowance has to be made for the offer of the Kingdom to the nation of Israel. If Israel had accepted the offer of the Kingdom that Jesus proclaimed, I think the destruction and seven year interval written in Daniel would have happened right away.
I think we see an allowance for the possibility the Israel would not accept the kingdom at that time when Jesus makes the announcement of his ministry.

Luke 4:17-20  And there was delivered unto him the book of the prophet
  Esaias. And when he had opened the book, he found the place where it
  was written,   The Spirit of the Lord is upon me, because he hath
  anointed me to preach the gospel to the poor; he hath sent me to heal
  the brokenhearted, to preach deliverance to the captives, and
  recovering of sight to the blind, to set at liberty them that are
  bruised,  To preach the acceptable year of the Lord.   And he closed
  the book, and he gave it again to the minister, and sat down. And the
  eyes of all them that were in the synagogue were fastened on him.
Isaiah 61:1-6  The Spirit of the Lord GOD is upon me; because the LORD
  hath anointed me to preach good tidings unto the meek; he hath sent me
  to bind up the brokenhearted, to proclaim liberty to the captives, and
  the opening of the prison to them that are bound;   To proclaim the
  acceptable year of the LORD, and the day of vengeance of our God; to
  comfort all that mourn;   To appoint unto them that mourn in Zion, to
  give unto them beauty for ashes, the oil of joy for mourning, the
  garment of praise for the spirit of heaviness; that they might be
  called trees of righteousness, the planting of the LORD, that he might
  be glorified.   And they shall build the old wastes, they shall raise
  up the former desolations, and they shall repair the waste cities, the
  desolations of many generations.  And strangers shall stand and feed
  your flocks, and the sons of the alien shall be your plowmen and your
  vinedressers.   But ye shall be named the Priests of the LORD: men
  shall call you the Ministers of our God: ye shall eat the riches of
  the Gentiles, and in their glory shall ye boast yourselves.

The portion of Isaiah that Jesus read did not include the full text. He stopped before the proclamation of the "day of vengeance".
It is at this point where allowance is made for the nation of Israel to accept the Kingdom or not. When Paul wrote Romans, I see this offer to Israel still open. 

Romans 11:11  I say then, Have they stumbled that they should fall?
  God forbid: but rather through their fall salvation is come unto the
  Gentiles, for to provoke them to jealousy.

I see a future time when a faithful remnant of Israel will receive the kingdom.

Revelation 1:7  Behold, he cometh with clouds; and every eye shall see
  him, and they also which pierced him: and all kindreds of the earth
  shall wail because of him. Even so, Amen. 
Isaiah 66:8  Who hath heard such a thing? who hath seen such things?
  Shall the earth be made to bring forth in one day? or shall a nation
  be born at once? for as soon as Zion travailed, she brought forth her
  children.

The variability of when Israel would receive the kingdom bears directly on what might be called the time of the gentiles. I see potentially it might have been as short as seven years.
Paul had hoped that seeing gentiles come to salvation and manifest the works of the Holy Spirit would be sufficient to provoke Israel to jealousy and become motivated to, as a nation, receive the offered kingdom.
The last thing the disciples asked Jesus was when the kingdom would be established.

Acts 1:6-7  When they therefore were come together, they asked of him,
  saying, Lord, wilt thou at this time restore again the kingdom to
  Israel?   And he said unto them, It is not for you to know the times
  or the seasons, which the Father hath put in his own power.

Jesus was describing how Jerusalem would be under the control of gentiles for a period of time. Paul was describing a period of time in which gentiles were brought ("come in") to salvation without the aid of a faithful Israel. 
In dispensational doctrine, when Christians are taken out of the world at the rapture, the world reverts to an Israel centered perspective and the seven year period Daniel wrote about starts. This period of time could have been as short as seven years. The future 1,000 year reign of Christ on earth from Jerusalem could have been already established for two thousand years.

Romans 11:12  Now if the fall of them be the riches of the world, and
  the diminishing of them the riches of the Gentiles; how much more
  their fulness?

From various verses in the Old Testament we can get a picture of what things will be like and could have been like with the "fulness" of Israel.
No war;

Isaiah 2:4  And he shall judge among the nations, and shall rebuke
  many people: and they shall beat their swords into plowshares, and
  their spears into pruninghooks: nation shall not lift up sword against
  nation, neither shall they learn war any more.

Long life;

Isaiah 65:20  There shall be no more thence an infant of days, nor an
  old man that hath not filled his days: for the child shall die an
  hundred years old; but the sinner being an hundred years old shall be
  accursed.

Nature less hostile;

Isaiah 11:6  The wolf also shall dwell with the lamb, and the leopard
  shall lie down with the kid; and the calf and the young lion and the
  fatling together; and a little child shall lead them.

While we are waiting for the time of the gentiles to end and the arrival of the kingdom, we can take comfort that our King is faithful and will accomplish all he has promised.

