'Twilight' refers to the vampire's world that Bella entered when she fall in love with Edward.
'New moon' refers to the darkest time of Bella's life when Edward left her, because the night is so dark when it's new moon.
'Eclipse' refers to what Bella used to think that Jacob is her sun but Edward overshadowed him, so he can't win over him.
'Breaking dawn' refers to the transformation of Bella's life from ordinary human to so powerful vampire that shines.
"Twilight" is one of the times of day which is considered most magical. "New Moon" is the darkest possible time. "Breaking Dawn" is the hope of her finally being a vampire, and the strength of her new power winning the final battle.
The book is named "twilight", "New Moon", "Eclipse" and "Breaking Dawn" because it has something to do with transformation of human bodies to supernatural bodies which usually occur during these times.
I think I read somewhere that Twilight was on a list of 'words with atmosphere' and I guess they went with a theme of sun/moon related words. It did sort of mirror Bella's story arc - as the story gets darker the sun/moon phases seem to get darker.
