The cover of Twilight has an apple. The apple is a biblical representation of the knowledge of good and evil. When Eve ate the forbidden apple in the Garden of Eden, she became aware of the presence of good and evil. Bella symbolically eats the apple and becomes aware of the existance of vampires, both the good ones (the Cullens) and the evil ones (James).
The cover of New Moon has a tulip losing a petal. This represents the loss that Bella suffers when Edward abandons her. The falling petal can also symbolize the tears she cried and the drop of blood from the paper cut that prompted Edward's decision to leave.
The cover of Eclipse has a ribbon that is almost (but not quite) ripped apart. This represents Bella's difficultly in severing her ties with the human world. Notice that the ribbon is almost cut but not quite. Bella is almost ready to leave her human life behind, but there are some attachments that she is not ready to let go (Jacob, family, sex).
The cover of Breaking Dawn has two chess pieces: a pawn in the background and a queen in the foreground. In chess, the pawn is the least significant piece, while the queen is the most significant (and most powerful). Bella is both chess pieces. She begins the book as the pawn, weak and easily overcome. She transforms into the queen, the most powerful force to stand against the Volturi.
This information can be found in The Twilight Saga: The Official Illustrated Guide. Incidentally, the cover of this book shows puzzle pieces, symbolizes the coming together of all four books from the saga.
Meyer, Stephanie. The Twilight Saga: The Official Illustrated Edition. New York, New York: Little, Brown and Company, 2011. 539.