The simplest way to explain this term is that a botnet is a collection (or network) of software robots which run in a distributed fashion automatically, typically via a large, coordinated effort through the use of compromised computers (often called zombies) that are running programs (worms, trojans, backdoors). These infected computers are controlled under a common command and control infrastructure - that just means that there is a sort of "master/slave" relationship where these infected computers can be accessed remotely to do almost anything - from stealing passwords to showing popups stuffed with advertising or paid links - the list is rather long.
There have been several instances whenn botnets have been caught. For those in the know, the current trend is to have botnets host other botnets - does that make your head explode or what?