That prophecy spoken by Jesus Christ is going to be fulfilled somewhere in the area of the Temple Mount in Jerusalem. But what exactly did He mean by "abomination of desolation"?
First, the location, as specified in the prophecy. Jerusalem is in Judea, and the "Holy Place" (i.e. the Temple Mount, seen in the link above) is in Jerusalem. Can we therefore be sure that this is the Place To Watch? Since the last Temple was destroyed by the Roman Army in 70 A.D., a new Temple (apart from the Christian meaning and application of the word Temple) must be constructed by the Jewish people before the "abomination of desolation" can occur in it.
Second, the time when it will occur. If we read through to the end of the prophecy, we find, "At that time the sign of the Son of Man will appear in the sky, and all the nations of the earth will mourn. They will see the Son of Man coming on the clouds of the sky, with power and great glory." (Matthew 24:30). This clearly describes the time of The Return Of Jesus Christ. That great event has not yet happened, so we know that the "abomination of desolation" hasn't happened yet either.
So what do we know so far? (1) The "abomination of desolation" will occur at some time in the future, (2) in a new Temple, (3) that will be constructed on, or near, the Temple Mount, (4) just before the time of the Return of Jesus Christ. There are many other direct Scriptural references that fully support these points.
But what of the "abomination of desolation" itself? What will it, or he, be?
In the time Between The Testaments, the Syrian king Antiochus Epiphanes erected a pagan altar right in the Temple in Jerusalem and used it to sacrifice a pig to the heathen god Jupiter Olympus. Some have described this as the "abomination of desolation." The only problem with that opinion is that the incident occurred in 167 B.C. - after the book of Daniel was written, yes, but over 200 years before Jesus Christ made His reference to it in a prophecy about a future event. Antiochus Epiphanes could not have been the "abomination of desolation" - although he was an object lesson of what it will be.
Others believe that the Roman destruction of the Temple in 70 A.D. fulfilled Jesus' prophecy, but His Return did not occur then as is specified in the prophecy. The Romans at that time could not have been the "abomination of desolation."
So what is it going to be? The answer is found in your Bible -
It will be a human who erects something in the Temple of God that will then have been built.
He will lead a military superpower that will have conquered Israel. He will also claim religious authority, and because of his miraculous powers, millions will believe and follow him.
He, and it, will defile the Temple just before the Return of Jesus Christ.
He will claim to be not just a god, but God. He will establish himself in God's Temple which he will claim as his own.
He will have miraculous powers, from Satan, that will enable him to deceive and conquer.
There is absolutely nothing more abominable in the sight of God than a mere human claiming to be a god, let alone the true God. It is the absolute, ultimate blasphemy.
Your Bible clearly states that there is coming a great dictator who is going to have tremendous, seemingly miraculous, powers - but not from God. This man, for some time, is going to be nearly undefeatable. Allied with his Great False Prophet, he is going to deceive and enslave millions of people (Revelation chapter 13 describes their relationship very well, including how the military dictator takes the superior position in religious matters). His power will not last very long, but while it does, he is going to be the most powerful man on earth.
That man, and the idol that he erects to himself in God's Temple, will be the abomination of desolation. See Birth Of A Superpower.
Fact Finder: How long will the great power of this dictator last?