There are more than one way to do a barbell squat. There are also two ways you can do a traditional back squat. All of these exercises are great for your glutes, quads, calves, and hamstrings, although one variation of the squat may affect an area or muscle group better than the other.
Some people will argue that a variation of a squat is better than the other -- I am going to tell you that all of these variations of squats are important. They'll hit your muscle groups in different ways, and you might as well be good at all of them, than just one of them. Lifting weight should also be about variety and trying new lifts out!
I'd also keep in mind that depending on your body's proportions, strengths, weaknesses, your squat may look a bit different.
But here is the best advice I can give: have control of your weight. Keep tight and solid, learn breathing methods, and stay balanced. You don't want to risk injury to protect your ego. Lower the weights and keep improving your technique.
These are my favorite sort of squats. The bar is positioned high above your traps. Your foot placement, or stance, will be closer than your low-bar squat. Because of this, you'll be able to bounce back up from the squat. Your back will appear more upward with the high-bar squat. High-bar squats are usually used for Olympic lifting. In powerlifting, the low-bar squat is the popular squat. However, when I was powerlifting, high-bar squats was my go-to squat.
For the low-bar squat, the barbell is positioned a bit lower on your back. Also, your stance would be more shoulder-width apart. These squats will hit your hamstrings more than the high-bar squat.
Front squats are very awesome for your quads. When you correctly perform this squat, you'll feel your quad muscles working hard. Expect awesome soreness in your quads the next day!
Be careful with the weights you use with this one. You don't want to go so heavy that you cannot control the weight. Also, your stance can vary. I typically use the same kind of stance as I would with high bar (more narrow). However, it depends on what feels most comfortable to you.
These are difficult! This takes a lot of balance along with upper-body strength.