Going by this analogy, one can imagine all kinds of servers that can be used on a network. File servers are a very common use. All files are stored on one or more servers. Being central, it is easier to manage the files, maintain versions, and manage back up and restore of files of the organization easily. As the storage needs grow you add to the storage of the server. Similarly there can be a fax server that communicates with the external world via fax. The clients simply send the fax to the server or receive one from it. There can be a communication server that lets all the clients communicate via VoIP, IM, and so on.
Organizations also typically have a proxy server that communicates with the Internet and the internal clients communicate with this proxy server to request services from the net to this server. Another class of servers is application servers. The main application, that is typically multi-user, runs on the server and clients are able to operate this application from their machines. Typical examples in this class would be a centralized PDF converter, a CAD application, a centralized time reporting application, and so on.