A computer virus is a small program (written by a bad guy) who lands in other programs body.. When you run your infected program, the virus moves to the RAM of your computer and starts his dirty work: Data destruction, hard disk format, system errors.. A virus is a parasite who propagate rapidly thanks to Internet. Each virus has his own signature ( a sort of DNA). Such signature allows other anti-virus programs to detect this famous virus in order to erase it..and that's why you should regularly update your anti-virus.
First of all, you have to get a clue about what you'll be facing... There is many kinds of virus and you have to recognize your enemy in order to defeat him !
Worms : able to spreat through networks (Internet, Intranet...)
Trojans : creating a rift in the system allowing its creator to penetrate through it and take control.
Logical bombs : a remote virus, can be activated after certain events.
We can classify viruses as following :
Boot virus : Infects the boot sector (MBR : Make Boot Record), this sector allows the execution of your operating system.
Polymorphe virus : it changes its appearance just like a chameleon. It's an intelligent virus which can take many forms and different signatures.
Macros virus : Infects the MS.office documents macros.. It lands in a simple Word or Excel and executes a simple code once you open the infected document. It affects windows files too. This kind of viruses is usually sent by mail in attached files, easy to recognize with their .VBS extension.
that was a brief introduction, if you want to get more information check the Wikipedia definition.