In computer terms, worms are really a subset of viruses, but they have the ability to replicate by themselves, they do not require a host file.
Simply put, viruses infect hosts, and worms infest systems. Often worms exploit a vulnerability in services in network facing services. Such worms can spread very quickly across networks of vulnerable systems, as they do not require any intervention from users to run. However, the commonest type of worms are carried in emails (it is important to note that it is not the email which is infected, but that they carry the worm files). In the case of the email borne worm, the recipient of the email is the vulnerability that is exploited, usually with an enticing subject or message.
Usually worms are much easier to remove from a system than viruses, because they do not infect files. Worms often try to add them selves to the startup folder, or modify registry keys to ensure that they are loaded every time the system starts. Again, worms do not necessarily have to do any damage (See payload).