Well, it's appropriate to discuss the actual meaning of the word "avatar" first because also has connections to the sacred. Avatar in the religion of Hinduism is the term for a deity that purposefully comes down to earth. The deities of Vishnu and Shiva come to mind automatically. In my opinion, this connects to the movie Avatar directly in that humans are, in a sense, making themselves unholy "gods" by choosing an "avatar" and invading the lives of the Na'vi. On a more positive note, the notion of an avatar connects to the movie because, through the help (the will?) of Eywa, the main character of Jake actually becomes his avatar.
This being said, the real sacred story in Avatar has to do with the Na'vi and their religion which focuses on the worship of the nature goddess named Eywa. It is a sign from this Eywa that originally convinces Neytiri to bring Jake back to her clan. We learn throughout the course of the story that the natural world, the Navi, and Eywa are inherently connected (hence the danger of them leaving Hometree to its destruction). A cornerstone of this link is the Tree of Souls where the Na'vi can communicate with various forms of indigenous life on Pandora. This connection is again apparent when the animal life of Pandora "defeat" the humans with the aid of the wounded Na'vi.