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Actually, the reason for the title of the movie, as well as some of the other answers were made quite clear
Why did his mom make it with an alien? she didn't Much like an old Twilight Zone, the boy had great powers and willed the alien into existence from his mind. However, he doesn't realize his powers. One night, he is watching an old alien movie titled "the Day the World Ended" - this explains the title, as well as the alien.
Notice the "alien" only kills those that killed his mother and then later vaporizes into thin air.
While the story was a little weak, there was some thoght required. The title was only referenced twice, and quickly, so it was easy to miss.

maybe this is like jeepers creepers and they want you as the audience to create the backstorys etc.
And You Shall Know My Name Is The Lord When I Lay My Vengence Upon Thee

I don't think the mother actually had any psychic power I think she was just covering for the son. Like when she said before they killed her something like "You have to stop doing this, don't hurt them again" I got the feeling that she wasn't just talking about burning them in the diner. She may not have been his real mother, he could have been an alien like she says, or some government experiment she hid away there (she looked even more out of place than the therapist) Who knows?
And then as far as the alien goes, it may have been a product of his power just like the adopted father said. It sure did look a lot like the one in the movie he watched, and his memories had come back to some extent right before it appeared, so it could have been a product of his subconscious desire for revenge.
Nothing was made terribly clear in the film, I guess and when I saw it the title was "Creature Features: The Day the World Ended", which makes me wonder if maybe they mixed the title up with some other Creature Features or something and didn't bother to change it.