There are almost as many different classes and sets of rules as there are contests. There is no standard, things vary with the part of the country that you are in, and from show to show. Most commonly, the classes are:
stock, either saw provided by the contest organizer, or saws brought by the contestants. Sometimes they check to see if the saws are stock, sometimes they don't, which leads to:
stock appearing, where the saws look stock but are highly modified;
super modified, where the saw had to start out as a chainsaw but can be modified with expansion chambers etc., altering the appearence of the saw;
All of these can be broken down into displacement classes. This is commonly done on the west coast, where they sometimes run a less than 6 cu. in. class and an over 6 cu. in class
open, or "motorcyle", where go cart, motorcycle or snow mobile engines are used, the only restrictions are that they be one cylinder and started with the pull of a cord.
unlimited, where car engines are used, it takes two people to lift one of them.
Alot depend on where you are. The "unlimiteds" are usually found in the pacific northwest. In New York, you may see a hotsaw class along with a stock class where the contestants bring their saw, often and 3120 or 088. In West Virginia, they prohibit "motorcycle" saws, and run only super modifieds. Across the Mason Dixon line in Pennsylvania, most, but not all, of the contests allow motorcyle saws. It is confusing, and you have to check the rules everywhere you go. I once showed up at at contest entered in the stocksaw class thinking the saw would be provided. It wasn't.
Your are usually safe if you show up with a big chainsaw which has been visibly modifed. Of course, if the motorcycle guys show up, you will get smoked.