Hey now, I'm the expert on chatting about wrestling.
There are two international styles of sport wrestling. One is Freestyle (FS) and the other is Greco-Roman (GR). They are essentially the same techniques, with one difference, in GR you cannot grab the legs, use the legs to trip or tackle the opponent, or secure the body with the legs. GR is sometimes called 'classic' style, because it is believed to be similar to ancient greek wrestling (which isn't exactly true).
In freestyle (FS) you can use both the upper and lower body to attack and defend. That means you can tackle and trip as well as do an upper-body throw.
Greco focuses on throws and wrestling in the standing position, freestyle focuses on 'shooting' or 'tackling' and the counters to tackles. In Greco, people stand more upright, since there is no danger of being tackled, and in freestyle they crouch more, and tie up with the arms to defend against being taken down.
American Folkstyle, or Collegiate, or Scholastic, Wrestling is what I consider the best style and is a uniquely american sport. It is essentially like freestyle, because both upper and lower body takedowns are allowed, but the mat wrestling is different.
In Collegiate, the top man gets points for 'riding' or being on top, the more time, the more points. The bottom man can get points for escaping to neutral (standing up facing the opponent) or reversing (going from bottom to top). This point system encourages very active wrestling on the mat, as opposed to FS and GR, where the defensive (bottom) wrestler basically spread his arms and legs and waits for the referee to intervene. In FS and GR, you are only allowed about 15-20 seconds of mat wrestling, before the referee stands you up again.
So in summary,Greco Roman is upperbody wrestling, freestyle is lower body wrestling, and American Folkstyle (collegiate) is a variant of freestyle where mat wrestling is emphasized.
I need some iced coffee now, I'm exhausted.