When it comes to powerlifting, a large portion of lifters will use a low bar squat which places the bar lower on the back to sit on a muscular shelf made by the rear deltoids. This also allows the lifter to have more forward torso lean throughout the squat and shifts some of the tension to the posterior chain. This together tends to allow a lifter to lift more weight maximally. However, low bar squatting is not a requirement for powerlifting; high bar squats (with the bar on the traps) is also perfectly valid. For powerlifting, which ever allows the lifter to lift more weight during competition is the better variation for them.
For hypertrophy, high bar squats might be more effective to a degree as it places more demands on the quads than low bar squats. This doesn't mean that low bar squats are bad for hypertrophy either. It really comes down to which is more comfortable to use and the particular rep scheme you're using.
- Box: a variation in which a bench or other sturdy device is placed around where you'd squat to parallel. The purpose is to squat down and touch the bench (or box, etc) and come back up, or pause slightly before coming back up. It is a tool to help with both technique and to help with driving out of the hole, since there is no stretch reflex available in this variation.
- Sumo: a variation with a very wide stance. This will shift some of the load onto the hips and adductors. This will allow for a shorter range of motion that still allows you to squat to parallel. Lots of geared powerlifters appear to use this kind of squat.
- Front: in this variation, the bar is racked across the front of the body in the channel created by the front deltoids and clavicle. This variation places more demands on core stability, as you need to maintain a very upright torso position to avoid dumping the bar. This also causes more demand on the quads. This is also one of the most important accessory lifts for Olympic weightlifting, as it is the second half of a clean.
- Hack: you can think of these as deadlifts with the bar behind you. In a normal deadlift, the hip tends to be high enough that the quads contribute little compared to the hamstrings and glutes. In a hack squat, it's much easier to get into a lower hip position which will allow the quads to play a larger role in lifting the weight up.
- Overhead: for most people, this variation only serves as a way to help with core stability, since you need to stabilize the barbell overhead while also squatting with it. If you have tight shoulders, avoid this movement until you can comfortably hold the bar overhead in a snatch grip with a slight forward lean of the torso. Olympic lifters also use this variation to help strengthen the squat portion of the snatch (the bar is caught in an overhead squat position in a snatch). They also use it for mobility and technique training.