Use of the Internet involves pairs of computers exchanging messages. Generally, some computers are more popular than others; they don't all talk to each other equally. The more poular ones, called servers, also generally don't initiate conversations, except maybe with other servers. The less popular ones, and the ones you're more likely to actually see, are clients. This is known as the client-server model, and it's fundamental to the Web and most other applications of the Internet. (Computers are sometimes more egalitarian, and then they're known as peers and engage in peer-to-peer (P2P) interaction.)
The software which causes a computer to act as a client or server is also known by the same term. So, for example, Web browsers are a type of Web client, GitHub runs (one or more) git servers on its physical servers, and Visual Studio comes packaged with a Web server.