The 'client' is used in client-server environment to refer to the program the person uses to run the application. A client-server environment is one in which multiple computers share information such as entering information into a database. The 'client' is the application used to enter the information, and the 'server' is the application used to store the information.
A web application relieves the developer of the responsibility of building a client for a specific type of computer or a specific operating system. Since the client runs in a web browser, the user could be using an IBM-compatible or a Mac. They can be running Windows XP or Windows Vista. They can even be using Internet Explorer or Firefox, though some applications require a specific web browser.