Twilight, the apple on the cover of Twilight represents "forbidden fruit." S.M used the scripture from Genesis (located just after the table of contents) because she loved the phrase "the fruit of the knowledge of good and evil." Isn't this exactly what Bella ends up with? A working knowledge of what good is, and what evil is. The nice thing about the apple is it has so many symbolic roots. You've got the apple in Snow White, one bite and you're frozen forever in a state of not-quite-death... Then you have Paris and the golden apple in Greek mythology—look how much trouble that started. Apples are quite the versatile fruit. In the end, she loved the beautiful simplicity of the picture. To S.M it says: choice.

New Moon, the ruffled tulip. S.M helped choose the apple for Twilight, but had no control over New Moon's cover. She doesn't know what it's supposed to mean.

Eclipse, the torn ribbon. This isn't on S.M's webpage like the others so I've put together a few answers. It could be Edward over shadowing Jake, which happens in the story. It could symbolize Bella + Jake's relationship being torn apart by Edward for Bella's safety. Bella has to choose who to be a permanent fixture in her life or Bella trying to break away from her life to be a vampire.

Breaking Dawn, the chess pieces. SM said that the red pawn represents Bella as a human. She was pretty much, metaphorically, the weakest player on the chess board. The white queen, represents Bella as she transforms from the weakest player to the strongest one; (her change from a human into a vampire).
SM also said that the cover foreshadows the volturi's confrontation at the end of the book. It foreshadows that the volturi confrontation is merely a court scene, and not a fight scene.