For NPN/PNP, think of it like a lever, or hydraulics. With a certain current (voltage and resistance) at the base, you get a proportionately lower resistance between emitter and collector, and thus a proportionately higher current through them. In this way, it "amplifies" current by a set factor, known as the "gain". Bipolar transistors are relatively linear, in that as you raise the base voltage (thereby sinking more current through the base), they just keep conducting more and more. At some point, they won't improve much. This is called "saturation" and is where they are for most purposes, fully on.
For MOSFETS, it's voltage controlled. Not immediately, but at a certain voltage on the gate, the drain begins to conduct to the source. At a certain voltage beyond that, the transistor is "fully on" and further input voltage realizes little to no improvement in the decreasing resistance between drain and source. So, it acts like a voltage-controlled resistance up to the point where it acts like a switch. Unlike NPN/PNP, the gate draws no appreciable current. (Although, the gate will have a certain amount of capacitance that will draw current until "full", but that is a short transient event when voltage is first applied to the gate, not a continuous load.)