Babylonia was the state that grew up around the city of Babylon.
The Empire that is usually thought of when Babylon is mentioned is the Neo-Babylonian or Chaldean Empire. A revolt in 625 B.C. against the rule of the Assyrian Empire saw the destruction of the Assyrian capital Nineveh in 612 B.C. Carving up the Assyrian Empire between themselves and the King of Media who had also revolted, a new Babylonian Empire was born which quickly rose to military prominence until its defeat by Persia in 538 B.C.