The movie is about an alien who accidentally gets left behind on earth and subsequently forms a close relationship with a boy named Elliott. Elliott gives the alien the name E.T. (for "extra terrestrial").
Throughout the movie, there are many instances where their close relationship is illustrated. Shortly after they meet, E.T. mimics Elliott's movements as they get to know each other. In one scene (even called "A Psychic Link" on the chapter menu on the DVD), while Elliott is at school, he seems to experience what E.T. is doing at home. Later, when Elliott's brother Michael comments that E.T. "doesn't look too good anymore," Elliott brushes this off with, "Don't say that; we're fine!" and Michael replies, "What's all this 'we' stuff? You say, 'We' all the time now." When government officials discover E.T. and invade the family's house, asking probing questions, Michael tries to explain that "Elliott feels his feelings," and one scientist comments, "EEG analysis shows complete coherence and synchronization of brain wave activity between both subjects."
I realized that, in a small way, their names point to this sort of symbiosis, too: both start and end with the same letters.
I also noticed a couple small references. First, the DVD cover, which is apparently one of the original posters for the movie, bears some resemblance to a detail in The Creation of Adam, part of Michelangelo's painting on the ceiling of the Sistine Chapel.
Second, when E.T. sees a kid dressed up as Yoda for Halloween, there's a brief quotation in the soundtrack of Yoda's theme from The Empire Strikes Back. Both soundtracks were composed by John Williams.
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*In I'm Told I Had a Good Time: The Micky Dolenz Archives, Volume One (page 429), Dolenz comments that Allen Daviau, who was the cameraman for the production company that he started in 1975, was later the cinematographer for E.T. In the movie's credits, he's listed as "Director of Photography."