The "server" is the computer that "serves" data or services.
The "client" is the computer that asks for the service or data from the server.
Interestingly, the World Wide Web is probably one of the best examples of a client/server architecture. When accessing the World Wide Web, you use client software (usually Internet Explorer or Mozilla Firefox), which connects to the web server, which holds the content. The web browser asks for the content, and the web server serves it back to the browser.