To put it another way:
A server is a computer that provides servies (hmmm) to other computers on a network. File servers store files, print servers expedite network printing, web servers allow access to web pages, etc... Physically, servers in a business environment are frequently just beefed up takes on desktop computers, using the higher end of similar technology.
Mainframes are very large, very expensive supercomputers that were used "back in the day" (and are still used by companies who haven't replaced/upgraded them yet) as servers. Desktop-type technology wasn't powerful enough to provide services for a large-scale network, so these larger, more powerful computers were necessary. They're more or less obsolete now, as the same services can be provided at lower cost by one or more modern servers.
I think that's a more or less accurate, but broad, explaination... Someone correct me if I've missed anything important...