This is Turul statue at the gates of the Buda Palace.

The Turul is the most important bird in the origin myth of the Magyars (Hungarian people). It is a mythological bird, a divine messenger, and perches on top of the tree of life along with the other spirits of unborn children in the form of birds. The Turul became a symbol of power, strength, and nobility of the Hungarian people, and it is still used today on the coat of arms of the Hungarian Army and the Office of National Security. (from Wikipedia)
http://en.wikipedia.org/wiki/Turul