Is the relationship between Enkidu and Gilgamesh an important element of the story?

The friendship between Enkidu and Gilgamesh is the story. They begin as enemies, since Enkidu was created by the gods to counter Gilgamesh's "rape the women and kill the men" style of ruling his kingdom, but after they fight almost to the death, they accept each other's strength and determination and become very close, loyal friends. They basically get into a bunch of shenanigans together to prove their manliness, and because of one stupid move, Enkidu is cursed and ends up dying. Gilgamesh mourns him for days, and after he buries him he becomes a wanderer (the complete opposite of his previous kingly lifestyle). He travels the world alone, humbled and destroyed by the loss of his friend, literally contemplating the meaning of life. He eventually hears of a plant that will give eternal life, though almost immediately upon finding it he loses it forever, showing just how much fate plays into our lives. After Gilgamesh accepts the role of fate, he dies and everyone honors him for the changes he made in the way he lived his life after becoming friends with Enkidu.

I hope that helped a bit, though I definitely recommend actually reading the epic. It really is a spectacular read. Good luck in whatever you needed this for!