A server is a running instance of an application (software) capable of accepting requests from the client and giving responses accordingly. Servers can run on any computer including dedicated computers, which individually are also often referred to as "the server".
Servers operate within a client-server architecture. Servers are computer programs running to serve the requests of other programs, the clients. Thus, the server performs some tasks on behalf of clients. It facilitates the clients to share data, information or any hardware and software resources. The clients typically connect to the server through the network but may run on the same computer. In the context of Internet Protocol (IP) networking, a server is a program that operates as a socket listener.
Servers often provide essential services across a network, either to private users inside a large organization or to public users via the Internet. Typical computing servers are database server, file server, mail server, print server, web server, gaming server, and application server.
Numerous systems use this client server networking model including Web sites and email services. An alternative model, peer-to-peer networking enables all computers to act as either a server or client as needed.
Basically speaking, if you need to run sever computers, that need to access the same database, all the computers can be networked to one server with several terrabytes of storage space and everyone connected can access that data at the same time. Same goes for programs. One program can be run through the server and every computer networked to it can access that software. There are many other uses for servers, such as a home server for instance. That can be used to store all the media that each computer or laptop or tablet on the same network can access. Store all of your movies, music and photo's on the server and everyone in the home can watch different movies or listen to different music at the same time.
Think of it as a bank, you store your important stuff on it, and you can take it out when you need to, without the hassle of losing storage space on your computers. Pretty good deal.