The Enuma Elish is the Babylonian Creation Myth. Written across seven stone tablets, the poem describes the beginning of the universe as a separating of water from Chaos. Essentially, the universe begins in a chaos of swirling waters until, with time, the waters separate into Apsu (fresh water) and Tiamat (salt water.) Apsu and Tiamat give birth to the first generation of gods, the oldest among them being Ea (sometimes Enki). You might say it's a water birth. (Yeah, we went there.)
Unfortunately, Ea and his younger brothers are noisy and disruptive, and they bug Apsu as he's sleeping. So Apsu wakes up on the wrong side of the bed and decides that the solution to the problem is to kill everyone. Tiamat intervenes by warning Ea, who is then somehow able to lure Apsu back to sleep and murder him. Hey, at least Uranus only got castrated. Or maybe being castrated is worse than death, it's hard to say.
In any case, Tiamat is outraged by the murder of her husband – sort of unfair after she went to the trouble of warning Ea – and she declares war on the entire group of younger gods. In hindsight maybe Tiamat should have given the matter more thought, because Ea promptly gives birth to Marduk, who then splits Tiamat in half with an arrow. Tiamat's blood and bones dissolve to create the earth and all of its creatures.
Reading even briefly over this story we're forced to admit that there are a lot of similarities between this and the events in the Theogony. Let's break it down mathematically.
Apsu = Uranus
Tiamat = Gaia
Ea = Cronus
Marduk = Zeus
The only major differences are that Zeus dethrones his father, Cronus, where Marduk defeats his grandmother, Tiamat. Oh, and in the Theogony, nobody dies. The Babylonians were less forgiving, apparently; or they hadn't built any good prisons yet.
It's difficult, if not impossible, to determine whether or not the Enuma Elish had any direct impact on ancient Greek beliefs about the origins of the universe, but given that surviving tablets containing the Enuma Elish found at the library in Ninevah date to about 1100 BC, we can at least say that the Babylonian story is older than Hesiod by about 300 years. What's more, many scholars believe that the Enuma Elish contains pieces of an even older story, from Mesopotamia. This is the sort of thing that makes it so difficult to sort out the underlying truth in ancient mythology.
If you want to know more about the Enuma Elish, be our guest.