A downloadable virus may be contained in a larger program such as a game, and it will replicate itself when the program is run. The virus may find other programs on a person's computer in which it could store a copy of itself. It is difficult to know whether a downloaded file or program contains a virus until problems arise on the computer such as weird messages, programs running that shouldn't be, the computer running incredibly slow, or files missing. Often, a virus scanner program will detect the virus, but not always.
On occasion trojan horses have been downloaded unwittingly by people. Although they have no way to automatically replicate, these viruses tend to be quite difficult to get rid of. The programs will generally damage your computer when run (opened).