A client is a computer system that accesses a (remote) service on another computer by some kind of network. The term was first applied to devices that were not capable of running their own stand-alone programs, but could interact with remote computers via a network. These dumb terminals were clients of the time-sharing mainframe computer.
The client-server model is still used today on the Internet, where a user may connect to a service operating on a remote system through the internet protocol suite. Web browsers are clients that connect to web servers and retrieve web pages for display. Most people use e-mail clients to retrieve their e-mail from their internet service provider's mail storage servers. Online chat uses a variety of clients, which vary depending on the chat protocol being used.
Increasingly, existing large client applications are being switched to websites, making the browser a sort of universal client. This avoids the hassle of downloading a large piece of software onto any computer you want to use the application on. An example of this is the rise of webmail.