Situated in The Tigris-Euphrates Valley, Babylonia has been referred to as "the cradle of the human race" for somewhat good reason - the Garden of Eden was located somewhere in it (see The Seven Days Of Creation and Where Was The Garden Of Eden? and The Creation Of Adam And Eve).
The Old Babylonian Kingdom was at its peak at about the time of God's calling of Abraham (see Abraham, Man Of Faith), who was from Ur Of The Chaldees. However, after many centuries of conflict, the old empire eventually became subject to the Assyrians, from about 885 to 607 B.C. It was during that period that the Assyrians (see Ancient Empires - Assyria) conquered and took into captivity the northern kingdom of Israel, from which the "Lost Ten Tribes" never returned (2 Kings 17:1-23).
Babylonia was divided into Accad to the north, and Summer ("Shinar" of the Old Testament) to the south. Along with Ur and the city of Babylon itself, other major cities were Uruk, or Erech (Genesis 10:10), Larsa, or Ellasar (Genesis 14:1), Sepharvaim (2 Kings 17:24), Eridu, and Calneh (Genesis 10:10).
The New Babylonian Empire, which existed from 606 to 536 B.C., fully conquered the southern kingdom of Judah in 586 B.C. It was then that the Babylonians under the rule of King Nebuchadnezzar completely devastated the city of Jerusalem, looted and burned the original Temple of God, built by Solomon (see Temple Mount Treasures), and carried the people of Judah, including the prophets Daniel and Ezekiel, off into captivity.
In 536 B.C., after 70 years of supremacy, the Babylonian empire, the "head of gold" in Daniel's Statue, came to an end when it fell to the Persians (see Ancient Empires - Persia). There is much more yet to come however - "Babylon" has many prophetic applications that have yet to be completed (Revelation 18:1-24).
Fact Finder: Does God refer to a great apostate end-time church as "Babylon the Great"?