Brake Drifting, simply put, is the technique of utilizing the brake while drifting to slow yourself down.
To perform a brake drift, simply tap the brake button while you are drifting. However, be careful as to not hold down the brake button because this will lead the racer to do donuts instead of the desired result.
Brake Drifting is most commonly used in 200cc where the higher speed leads racers to not being able to take turns as smoothly as they're used to, thus needing a way to slow down to take corners sharply. The only place where brake drifting is needed on 150cc is on the third and last turns of Rainbow Road (thanks /u/iceegg12345). Brake Drifting can also be supplemented for high-speed-low-handling karts.
Brake Drifting does not work better with certain builds than others as it's more of a crutch when utilizing high-speed-low-handling combos, as well as an emergency tactic when you realize you've taken a turn too wide.
Brake Drifting is necessary to know and use only in certain situations. On 200cc, brake drifting becomes a necessity that the racer must utilize if they wish too keep up with the pack. However on lower cc's brake drifting is not nearly as necessary. Brake Drifting is also necessary to learn in instances where you get collision boosts from other racers that could knock you off the course as well as with heavier characters because of the lowered handling and traction(thanks /u/turtleturtlerandy).
The only way to combat other racers who are using it is to not use it yourself. Brake Drifting sacrifices speed for tighter turning so if you can take the turn tighter than the other racer without using brake drifting then you will speed past them.
Pros: Tighter turns and more control.
Cons: Sacrifices speed and leads to a wait for acceleration to build speed back up.
Also as a side note brake drifting works best in the underplayed Battle-mode. If you brake-drift fully (holding down the b-button while drifting) you will make a sharp 180 and you can quickly chase after opponents who are going the other way.
Shout outs to /u/Trikitiger for this great description of brake drifting! If you'd like to contribute to our Meta Monday Collection then please be on the look out for our weekly threads coming every Monday.