What is the desolating sacrilege spoken of by the Prophet Daniel standing in the Holy Place?

Answer 1

It refers to the idolatrous image that was placed at the site of the Second Temple after the Temple was destroyed (Talmud, Ta'anit 28b). See also:

More about Daniel

Answer 2

A Christian viewpoint: Jesus mentioned this prophecy at Matthew 24:15,16 and said when his followers see the "abomination of desolation, spoken of by Daniel the prophet, stand in the holy place, then let them which be in Judaea flee into the mountains".

The original fulfillment of this prophecy was in 66 CE, when the Romans invaded Jerusalem, including the temple. The Roman army had all but won and were standing inside the temple with their army standards, but did not destroy Jerusalem at that time, and they unexpectedly left the city. This allowed the people that listened to the warning of Jesus to flee the city and go into the mountains.

Four years later, in 70 CE, the Romans came back and completely devastated Jerusalem and destroyed the Temple. The people who listened to Christ's warning, and stayed away from the city, were spared the horrors that occurred during this second invasion.

In the duality of prophecy, and the sign of the end, the "abomination of desolation" which is standing in a "holy place" is considered to be time when the corrupt governments of mankind, at some point and in some way, attack the religions (the figurative holy place) of the world.