That depends on the virus. Some don’t do much at all or they do not work as the author intended. Seriously, there are viruses that are as in poor quality as the software they attempt to take advantage of. Other viruses will wipe everything off of your computer. Some will make your computer little more than a drone responding to commands by whoever distributed the virus. Others yet will delete certain files on your system or install what is know as spyware in order to generate revenue off of you each time you use your computer. They might generate popup ads or move the cursor in the oposite direction when you try to use the mouse.
The Virus Encyclopedia at SARC has most viral effects mapped out pretty well, but they have only recently started including other malware (like spyware, for example) in their encyclopedia. If you know which virus you have, look it up there. Depending on what antivirus software ‘discovered’ the virus on your computer, it may not be using the same popular name as the one SARC uses for the virus. Don’t worry, most viruses are listed to include all the other vendors’ names for each virus as well.
If you are not infected, but want to find out how powerful a virus can be – you should check out the encyclopedia, too.
It is entirely possible that the virus could limit what you see on your screen, how the data in your files is presented, how and if your computer is capable of printing, and some can delete or respond to email automatically (and you will never see the email). Some viruses actually change the layout of windows, or infect the font cache to set a really annoying or ugly font as the system default (has anyone ever seen “critter”?).
Many older viruses have a trigger date which, usually on an annual basis, is the given date and time when the virus releases it’s payload. Some payloads can be avoided just by not having your computer turned on during this moment or on the targeted date. Others use a randomized trigger, which means that you have, for example, one chance in ten that the next time you reboot your computer the virus will delete all of your picture files.
To summarize: viruses can do nearly anything on your computer you can do, and much more, much faster. Ideally, if you can prevent yourself from getting infected in the first place, you are far better off.