Folkstyle (what Remy describes as high school) is a rounded type of wrestling, where a few points are awarded for a few events in a match, but for the most part the goal is to pin the other wrestler. Highlights of this ruleset are a two-second pin count, many types of illegal holds and maneuvers, and ample time limits.
Greco-Roman is the older style of wrestling, what is usually perceived as Olympic wrestling. One of the main goals (at least in the Grecian times) is to get your opponent off the ground. This style is heavy into throws and makes allowances for starting on your feet more often.
Freestyle rules make it easier to get a pin (0-second pin count, if the shoulders touch, it's a pin), and also awards more types of maneuvers points. Just being in control and rolling your opponent 360 degrees gains you a point. The focus is on holds and on-the-mat wrestling, as opposed to Greco-Roman style. More thought and reflexes are required, and the match tends to move quicker.
Collegic wrestling adds the idea of gaining points for staying in control of the match, called "ride time". Any time a wrestler is in position that prevents escape or reversal, the clock is running in his favor. At the end of the match, if he has more than a minute of control time, he is awarded one point.
Contrary to what most people will say, wrestling is not a sport. You don't "play" wrestling. It is a lifestyle. You live it, eat it, breathe it. I trust completely my teammates from high school, and they trust me. It teaches you that you can overcome anything or anyone through hard work and determination, but also that there is always someone better than you. Even though it is a 'team' event, it is one on one. No one can help you, and it's all or nothing at all.