A MOSFET is a type of *transistor*. Transistors are electrical devices which allow a small electrical signal to control a larger electrical signal. You can imagine them like electricity-controlled switches or dials, but with no moving parts.
MOSFETs are a specific kind, made in part out of metal reacted with oxygen (Metal Oxide Semiconductor). The transistor is controlled by a voltage inside of it creating an electric field (Field Effect Transistor). Metal Oxide Semiconductor Field Effect Transistor, MOSFET.