Here are the one’s I can think of:
Front Squat- Great for core stability and it forces you to keep your torso upright which is great for general squatting technique.
“Olympic” Back Squat or full back squat- A back squat where you go all the way down as deep as you can go (hamstrings on your calves). I think this hits the quads a lot and is good for flexibility and core work.
“Powerlifting” Back Squat- A back squat with a wider stance and generally only goes to about parallel. Usually is initiated more with the hips and thus hits the glutes a little more.
Box Squat- Somewhat of a variation of the powerlifting squat where you squat way back onto a box that is placed about parallel. You relax your hip flexors on the box for a second and then squat back up. It breaks the concentric-eccentric chain and is great for teaching to sit back and then activate the hip flexors. Really hits the glutes and hamstrings hard.
Hack Squat- I’m not a huge fan of this. Kind of like a deadlift with the bar behind the heels.
Overhead Squat- The king of stability and core exercises. You definitly won’t be able to use as much weight, but I don’t think you can find more of a true total body exercise.
Additionally, you can do squats with bands. This increases resistance as you get towards the top. Also, squats can be performed dynamically for a speed emphasis.