In net jargon (words people use about the Internet) a server is a computer that serves many things to user, or client machines. Usually a server will only do a few things. Every type of thing a server does is called a service. Services are used by other computers that are called clients. The relationship between the client and the server is called a client-server relationship. For example, Wikipedia has a web server which has a service for sending web pages over the Internet. Your client computer talks to the Wikipedia's web page service to get web pages for you. A server can also host internet games and allow file sharing.
So that servers and clients can talk to each other, they need to be connected to a network, and they need to use the same protocol. In networks, a protocol is a set way for machines to talk to other machines - like a language. For example, the Wikipedia server runs the HTTP (Hyper-text transfer protocol) to send web sites to your computer, and your computer uses the HTTP Protocol to ask Wikipedia for pages.
Usually, servers are specially made to be more powerful and reliable, and they are more expensive than normal computers. Sometimes, servers can be clustered, which means that lots of servers work together to do one service.
The server might slow down if there are too many people accessing the server at the same time, resulting in a high load. It might also be unable to take the high load, resulting in a shut down.
In a peer-to-peer system, every computer is both a client and a server to each other. This is commonly implemented in file sharing and VOIP. However, this can aid in attempts at piracy.