Yours, Etc.: Origins and Uses of 8 Sign-Offs
late 14c., from Anglo-French clyent (c.1300), from Latin clientem (nominative cliens) "follower, retainer," perhaps a variant of present participle of cluere "listen, follow, obey" (see listen); or, more likely, from clinare "to incline, bend," from suffixed form of PIE root *klei- "to lean" (see lean (v.)).
The ground sense apparently is of one who leans on another for protection. In ancient Rome, a plebian under protection of a patrician (called patronus in this relationship; see patron); in English originally "a lawyer's customer," by c.1600 extended to any customer.
A computer system or process that requests a service of another computer system or process (a "server") using some kind of protocol and accepts the server's responses. A client is part of a client-server software architecture.
For example, a workstation requesting the contents of a file from a file server is a client of the file server.