The English word "Babylon" comes from the Ancient Greek Babylon (Βαβυλων) which itself is a Greek transliteration of the Akkadian Babili(m).
Babili(m) probably means "gate of (the) god(s)", which is probably parallel to the Biblical narrative in which the ancient people of Shinar/Babylonia tried to "enter" Heaven by building a tower that could reach the Heavens.
However, the Bible offers a different etymology. Since the LORD didn't want this project to come into place, He confused their languages, as they all spoke one language at that time, and so it was called Babel (From Hebrew בבל translit. "Bavel"), which comes from the word Bilbel (בלבל), which in turn, is a verb form of the word Balal (בלל), which means "Confusion" or "Puzzle/Puzzling"
The city of Babylon began as the city of Babel, which was given its name, after God had scattered a crowded-together humanity and dispersed them across the earth by confusing their languages. It was founded by Nimrod who began many other cities in the region of Shinar, the area in which Babel was being built. At one point Shinar was later renamed "Babylonia" to reflect the influence of the city on the surrounding area.
The city of Babel, was constructed out of bricks instead of stone, and tar rather than mortar, which were different construction materials than most buildings of the time. In Babel a Tower was being constructed, for the purpose of giving honor to the city's builders, and to keep the people from spreading across the Earth.
At one point the city was renamed from "Babel" (referring to confusion of languages) to Babylon.
Babylon was the subject of many prophecies in the Bible, including some by Jeremiah and Isaiah. Babylon is prophesied by Jeremiah that it would rule over the land of Judah for a period of 70 years. He also said that this subjugation would include an exile that would end at the end of that 70-year period.
The neo-Babylonian empire rose to power in 612 BC with the defeat of the Assyrians at Nineveh. A few years later, in 609 BC, the Babylonians captured and killed the last Assyrian king. Seventy years later, in 539 BC, the neo-Babylonian empire fell to Cyrus and his invading army of Medes and Persians.