A virus is a computer 'disease' transmitted via the internet that can do irreparable damage to your computer in a very short space of time. The most common types are worms and trojan horses.
Worms spread from computer to computer and have the ability to replicate via your email client. They can send themselves to everyone in your email contact list – hence they can spread themselves very rapidly.
Trojan horses are generally mistaken to be useful as people think they are receiving official software or files from legitimate sources. In fact, they can cause incredible damage. Trojans at worst can create a 'backdoor' to your PC from the internet, for instance, forcing you to dial a premium rate number without you realising it. This type of covert attack is becoming prevalent with the growth of 'always on' broadband.