Transistors can be used in two main ways that are analogous to more common physical devices where a smaller signal (either voltage or current) controls a larger amount of voltage or current. The first is like a valve, were the smaller signal controls how much of a larger signal is allowed into the rest of the circuit. The second is like an on/off switch, where the presence of the small signal determines if the larger signal is completely blocked or not.
A "MOSFET switch" refers to a MOSFET, a type of transistor that is controled by voltage levels, configured to be such a switch.