Music server

A server is a computer somewhere in a network. There is a program ( the server process) listening for clients. If the client ask for something, the server will starts to do what the client  asked for (that’s serving).

A music server is a computer where the music is stored and has a process running which enables clients to browse its content and to stream selected songs to the client.

Any computer with the appropriate software can do the job but often a NAS is preferred simply because it is a small computer so low on power consumption (you don't need much performance to stream audio).
The music server software is often UPnP compliant.

Music server is also used to label a specific type of computers.
They are what they are, a computer with a soundcard, but they do have the look of audio gear.