MOSFETs are voltage gain devices with characteristics closer to vacuum tubes than BJTs. MOSFETs have a metallized gate separated from a semiconductor channel by a thin insulating layer. Charge applied to the gate controls the flow of electrons through the channel.
MOSFET .vs. BJT .vs. tube amplifiers is a subject of holy wars. I am firmly in the MOSFET camp. Typically, when MOSFET or tube amps distort, they enter a gradually increasing nonlinear mode, resulting in mostly sweet-sounding even order harmonics, whereas when BJT amps distort it is usually hard clipping, which results in harsh odd order harmonics. MOSFETS, unlike tubes, don't need periodic replacement, or ring when jarred.
Having said that, you are probably more interested in headphone amps. MOSFET and tube headphone amps have a fair amount in common, although many tube amps intentionally create even order harmonic distortion to sweeten the sound. Opamp based amps are all BJT output based, but really, opamps are in a class by themselves due to the peculiar nature of how they work.
An ideal opamp has infinite input impedance and gain, zero output impedance, zero DC offset, and probably some other ridiculous specs I don't recall, go read a book on it for more details. Anyway, opamps have a plus and minus input, and amplify the difference between the inputs. In order to get a "real world" gain, a feedback loop is incorporated using two resistors. Opamps are relatively easy to use and versatile, so they are common in electronics. We use them in headphone amps because of this convenience factor.
I hope this at least partially answers your questions.