What is the abomination of desolation?
The abomination of desolation is the idol image of the Antichrist that will be set up in the Holy of Holies of the rebuilt Jerusalem temple, demanding worship from the world and stopping the daily sacrifices.
Scripture teaches that in the middle of a seven year covenant the Antichrist makes with Israel, he will break that covenant, go into the temple in Jerusalem, sit on the mercy seat, and demand that the world worship him. Daniel 9:27 says 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. Daniel 11:31 adds that armies will stand on his part, pollute the sanctuary, take away the daily sacrifice, and place the abomination that maketh desolate there.
This abomination is identified as an actual image or idol of the Antichrist, set up in the Holy of Holies for the world, and especially the Jews, to worship. Revelation 13:11 to 15 describes a second beast who causes the earth to worship the first beast and gives power to an image of the beast so that it can even speak, and commands that whoever will not worship that image be killed. That image, placed in the temple, is what Jesus called the abomination of desolation in Matthew 24:15, quoting Daniel the prophet: 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.
Jesus made this the great turning point sign of the end times. The moment the abomination, this idol of the Antichrist, is seen standing in the holy place, it signals that the greatest and most severe tribulation the world has ever known is about to begin, so severe that those in Judea are told to flee immediately into the mountains without even stopping to gather belongings. This event marks the midpoint of the seven year covenant, ending the period of relative peace for Israel and opening the last three and a half years, the Great Tribulation itself, in which the Antichrist openly demands worship as God and turns against the Jews and the Church who refuse to submit to him.
Setting up this idol is described as the Antichrist's ultimate act of self exaltation, magnifying himself above every god and even against the God of gods, taking away the daily sacrifice and defiling the sanctuary that was meant for the worship of the true God. It is this desecration, the placing of the idol in the Holy of Holies, that is the abomination of desolation Jesus warned His followers to watch for.
The scriptures
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.
And arms shall stand on his part, and they shall pollute the sanctuary of strength, and shall take away the daily sacrifice, and they shall place the abomination that maketh desolate.
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:)
And I beheld another beast coming up out of the earth; and he had two horns like a lamb, and he spake as a dragon. And he exerciseth all the power of the first beast before him, and causeth the earth and them which dwell therein to worship the first beast, whose deadly wound was healed. And he doeth great wonders, so that he maketh fire come down from heaven on the earth in the sight of men, And deceiveth them that dwell on the earth by the means of those miracles which he had power to do in the sight of the beast; saying to them that dwell on the earth, that they should make an image to the beast, which had the wound by a sword, and did live. And he had power to give life unto the image of the beast, that the image of the beast should both speak, and cause that as many as would not worship the image of the beast should be killed.
And the king shall do according to his will; and he shall exalt himself, and magnify himself above every god, and shall speak marvellous things against the God of gods, and shall prosper till the indignation be accomplished: for that that is determined shall be done. Neither shall he regard the God of his fathers, nor the desire of women, nor regard any god: for he shall magnify himself above all. But in his estate shall he honour the God of forces: and a god whom his fathers knew not shall he honour with gold, and silver, and with precious stones, and pleasant things. Thus shall he do in the most strong holds with a strange god, whom he shall acknowledge and increase with glory: and he shall cause them to rule over many, and shall divide the land for gain.
From the books
This answer is drawn from these passages in the Mary Banks Faith Library.