I'l try to explain as best I can, but I'm not quite sure what your background is / how much you already know.
MOSFET is an acronym for metal–oxide–semiconductor field-effect transistor.
You can use a MOSFET in your circuit as an on and off switch. You would use the PWM output of your micro controller to control the speed of your motor by turning your motor on and off very quickly with the MOSFET, much as the motor controllers used in FRC.
Since your motor only spins in one direction, you only need a single MOSFET to control the motor. The motor controllers used in FRC use H-bridges, which are a combination of 4 transistors.
Here is an example of a single MOSFET driving a motor. By varying the duty cycle of the PWM signal, the motor's speed can be controlled.
EDIT:source -
http://www.picotech.com/applications/pwm_drivers/
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