Epic" is a Greek word applied to the two great narrative poems by Homer. When we apply it to other works, we imply that these other works have something important in common with the Homeric epics. In the case of Gilgamesh, the common elements are a strong central hero and a central concern with the relationship between human beings and death, the gods, and fate. Like Homer's poems, Gilgamesh also seems to be a central document of its culture. Unlike Homer's epics, Gilgamesh is a somewhat disconnected story: a collection of episodes in the career of the hero, rather than a single unified action.
Gilgamesh seems to have been a historical king of Uruk (a Sumerian city on the Euphrates) who, after his death, was turned into a legendary figure. (This often happens.) In the poem, he is presented as far larger and more powerful than anyone else, two-thirds god and one-third human. This is a problem for the citizens of Uruk, since he tends to bully them, rape all the women, and cause general misery in the city he rules. What to do? They call on the gods, who create a rival: Enkidu, a kind of savage half-beast who terrorizes the farmers. After an encounter with a prostitute, Enkidu loses his wildness, goes to Uruk, and wrestles with Gilgamesh. He loses, but they become best friends, and decide to seek adventures.