Metal Oxide Semiconductor Field Effect Transistor: A type of transistor that's suited to efficiently controlling large currents, hence commonly found on the business end of motor controllers, audio amplifiers, power supplies, etc.
MOSFETs are powered by magic smoke, and much of their specification pertains to the limits within which it won't escape. With something like a motor controller, unless they've really cocked up the design, the big question is usually thermal - how much heat the MOSFET will produce, and how hot it can be allowed to get without damage. How hot it gets for a given amount of power will depend on the physical design of the product it's installed in, and this is the bit that molishers of cheap Chinese kit will tend to take liberties with in order to keep costs down. Overspeccing the MOSFETs, bigger boards, chunky heatsinks, fans, thermal cutouts, etc all cost money (and may add weight/bulk, and make waterproofing more difficult), and skimping on them when "it'll probably be okay if they don't thrash it" is good enough for a product that's being marketed primarily on price.
For added excitement, these sorts of components tend to fail so that they're stuck 'on' rather than 'off', opening up the possibility of further expensive failures as a result. Fuses, of course, cost money and do nothing in normal operation...
Of course it might not be the MOSFETs. A controller for this type of brushless motor needs some sort of feedback to know the exact position the rotor's in, so as to apply power to the right coils at the right time. That may be done via Hall sensors inside the motor, which - along with the associated wiring - are relatively delicate and therefore another potential point of failure (think corrosion or mechanical damage).
And of course it's electronics, which means there's the usual power supply stuff to go wrong: Cheap nasty smoothing capacitors (which might not take kindly to being baked by the output MOSFETs) etc. If the power supply for the control logic goes out of spec, all bets are off.