The Enuma Elish account of the creation of the world is a religious, supernatural explanation for the natural phenomena of this current reality. The story follows the drama of the primordial gods, Abzu and Tiamat, the water serpents, who have always existed alone, making for a world without form, where primordial waters swirl about.
Then, the two accidentally conceive a whole pantheon of gods with different natures, which they consider to be evil, since they are discordant with the dominant elements of fresh water and salt water. The text follows the story of the created gods killing the two chaotic, primordial gods, first by overthrowing Abzu (with Tiamat's help). Then Tiamat becomes enraged and creates other beings, many of them serpents like herself, and waging war against the gods she helped to kill her husband. Eventually, a hero arises, the god of fire and lightning, and he electrifies her and kills her, dividing her body to create the world.
The literal narrative of the story, though, is only the beginning of its meaning. This text was the religious explanation of the existence of reality for the ancient Sumerians who lived in Babylon. Therefore, the analysis of the text should be done as religious exegesis (to draw the interpretation out of it).
One possible religious interpretation of the story might go like this:
Tiamat represents reality without order. Her ocean waters are a metaphor for the formlessness of the world. In order for the world to be created with its natural forces, there needed to be an opposite being to oppose her formlessness. When Marduk slays Tiamat, it can be seen as a religious statement against chaos. The Babylonians worship Marduk (and eventually his uncle, Ba'al) because they credit him to be the being who ordained the physical world and fought to create it.
One of the most useful analyses of this text would be a comparison and contrast to the well-known Hebrew creation myth.
They share many aspects. Both view the heavens as a realm where many beings are in contest for power. Both posit that the primordial state of reality was chaotic, formless waters. Both herald (G)ods who 'defeat the great serpent.' Both understand fire to be the dominant force of the universe. Both have a chain of being, where the gods exist above the titans and angels, who are above the animals, which are above the plants.
But they differ as well. The creation myth of the Hebrew people doesn't presuppose that their God was created out of chaos. In fact, this difference is exactly how the Hebrew people differentiated themselves to be unique. This has commonly been misunderstood to mean that the Hebrew people were monotheistic, when in actuality, they were probably henonists (believing in many gods, but only worshiping one).
The most interesting aspect of the Enuma Elish to a Judeo-Christian world is likely the fact that it includes gods who are mentioned by name in the Bible. The Jewish perspective is that the Babylonian gods are actually fallen beings who are to be destroyed.
These things, taken in harmony, help the reader to understand what the religious landscape was during the ancient Eastern days, long before modern Judaism, Christianity or Islam.