When the skies above were not yet named
Nor earth below pronounced by name,
Apsu, the first one, their begetter,
And maker Tiamat, who bore them all,
Had mixed their waters together,
But had not formed pastures, nor discovered reed-beds;
When yet no gods were manifest,
Nor names pronounced, nor destinies decreed,
Then gods were born within them. --Enuma Elish, tr. Stephanie Dalley
Apsu means freshwater abyss; Tiamat, salt water. The begetter and maker, father and mother.
I know some people, ranging from Robert Graves to Marija Gimbutas, have said that the story represents the conquest of an older matriarchal culture, represented by Tiamat, and its replacement by a patriarchal warrior one...but really, if that ever happened in Mesopotamia, it was already a done deal by the time the Enuma Elish was written, for a thousand years at least. We forget how long the Sumerians actually lasted; long enough to invent museums and the writing of history, after having invented writing in the first place. Those Sumerians had staying power. Then they got conquered by Babylon, who were basically a lot like Sumerians only a bit rowdier. Sort of like the British Empire fading out and the US becoming a world power, except the British Empire didn't last for hundreds and hundreds of years and we never actually invaded them, however it may feel to them during tourist season.
I digress. If you really want to understand what is going on in the Enuma Elish, you need to remember two things: floods, and civilization.
The word Mesopotamia means "between the rivers"...the Tigris and Euphrates, to be specific. The Sumerians and Babylonians lived in this twinned delta, near where the two rivers flowed into the sea. It was a fertile, and therefore rich and productive, place to live.
It also flooded a lot. A LOT. The Tigris and Euphrates were the ultimate source of the fertility, and therefore wealth, of the Fertile Crescent. Like Egypt and the Nile: no river, no crops, no civilization. Unlike the Nile though, which flooded in an orderly, regular fashion, the Tigris-Euphrates river valley flooded a lot more randomly and violently. Two rivers, after all, and different conditions. What made the Mesopotamians' lives possible also tended to kill them off fairly frequently. This accounts for the somewhat antagonistic, fatalistic and slightly morbid attitude the Mesopotamians had towards their gods.
"Tiamat our mother hath conceived a hatred for us."
Tiamat represents primordial chaos and deep water; she is also the mother of all the gods. She is the Creator, just as the regular flooding of the rivers literally created the Fertile Crescent. She is also a destroyer, and an unnervingly capricious one.
You can't have a civilization without a regular food supply. You also can't have one in the midst of swirling flood water all the time. It just doesn't work. Random chaos is why we can't have nice things.
Enter Marduk! He's a hero! He'll fix it! First Ea imprisons Apsu (= river levees?) then Marduk goes to battle and eventually slays Tiamat. He then proceeds to split her in half "like a fish" and create heaven and earth from the two halves of her body...imposing order on primordial chaos, shaping it into a fixed form. He then does something both interesting and revealing:
"He ordained the year and into sections he divided it;
For the twelve months he fixed three stars...
The Moon-god he caused to shine forth, the night he entrusted to him.
He appointed him, a being of the night, to determine the days (Long)"
Marduk invents time, and the regular division of time into years, months, weeks, days, and hours, marked by the movement of heavenly bodies. The seven day week and measurement of time in increments of twelve and sixty were in fact Sumerian inventions which we still use today, some five thousand years later.
Which brings us to the other thing: Civilization. The ancient Mesopotamians were quite conscious, and very proud, of their status as innovators and inventors of civilization, so much so that the major portion of the story of Inanna and the holy me consists of lists of godly powers (the me), which both constitute and create civilized, orderly life. The arts of writing, diplomacy, battle, and a few more scurrilous and racy arts all get a mention, as well as many pragmatic ones...bricklaying, for example, and counting stores. You've heard of people being house-proud; the Mesopotamians were civilization-proud. As well they might be. Most of their stories are about how awesome their civilization is, one way or another, including this one. Also the importance of constant vigilance...
Marduk slays Tiamat and imposes order on the universe. The ancient Mesopotamians drain the marshes, brave the floods and then invent everything including timekeeping. They build a mighty civilization, the influence of which still reverberates today. Order from chaos. Monsters and mud.