Amp rack is short for amplifier rack and is a term used mostly in reference to professional audio applications to describe any furniture, fixture, or case where amplifiers are mounted by their faceplates or in slot grooves, mostly synonymous with rack mount.
Examples include use in recording studios, mobile DJ setups, or live stage events. Each of these instances could require more power or more output channels than a single amp could provide, but necessitate portability of all amps at once.
The term is also sometimes used to describe the mounting fixture of an amp or amps used in car audio applications, though usually not the 19" rack mount variety.
References
Amp rack Wikipedia(Text) CC BY-SA