The "regular" squat
Position the bar on the trapezius muscles, not on the neck. Your head should be in line with your spine. Your chest should be up. Contract your abs. Your feet should be shoulder-width apart, and parallel or facing outwards slightly. Your knees should face in the same direction. Breathe in and bend your legs while pushing your glutes backwards, keeping your back straight. When your thighs are parallel to the ground, push on your legs while breathing out, to return to the starting position.
Benefits: Squats are great for strengthening the lower body, especially the quads and glutes. They work other muscles too, like the adductors, hamstrings, lower back and abs.
The full squat
The movement is exactly the same as a regular squat. The only difference is the range. Instead of stopping when the thighs are parallel to the ground, you go down as far as you are able to. Always remember to keep your back straight.
Benefits: A full squat exercises the glutes more effectively, and develops the muscles better in general, because of the increased range of movement and the number of muscles used.
The front squat
The movement is the same as for a regular squat. What's different is the starting position. The bar is not behind the head on the trapezius muscles but in front, on the upper pecs and front deltoids. You hold the bar with your hands in pronation and your elbows as high as possible so that the bar does not slide forward.
Benefits: Having the bar positioned in front accentuates work on the quads rather than the glutes. Whatsmore, the bust leans forward less, which means the back is supported better. So this exercise is perfect if you really want to feel your quads working, especially for people with long legs and a short torso.
The sumo squat
The movement is the same as for a regular squat except that the legs are much wider apart than shoulder-width and the toes are facing outwards.
Benefits: The sumo squat focuses more on the adductors (the inside of the thighs) than the regular squat. Also, the bust leans forward less, which works the back less and reduces the risk of injury to the lower back (this does depend however on your individual flexibility and mobility).