A worm is the type of malware that self-replicates; "malware" is short for "malicious software." A computer worm not only duplicates on one computer, but can also spread copies of itself over multiple computers that are connected or networked together in some way.
Worms often spread through e-mail, and many have mass-mailing capabilities that allow them to spread through a user's address book to many other users on other computers, which allows them to replicate quickly and dangerously. A worm is also a self-contained file, and it does not need to attach itself to an executable file.