Baggage size is measured in linear inches, which equals the length, width and height of a bag. For example, a bag that is 12 inches long, 10 inches high and 8 inches wide would be 30 linear inches. For flights within or between the United States, the U.S. Virgin Islands, Puerto Rico and Canada, American Airlines allows travelers to check bags that are up to 62 linear inches and 50 lbs; larger bags will incur a fine. For international flights, bags of 62 linear inches or smaller will not incur a fine, but the weight limits vary by destination. The American Airlines website has more information.
Carry on luggage must be no more than 45 linear inches. Passengers are allowed one bag plus a personal item of no more than 36 linear inches, such as a computer bag or purse. There is no weight restriction on carry-on luggage. Small items like coats and books do not count as personal items.
The TSA has imposed additional restrictions of what can actually go in luggage. There are differences between what can be put in checked luggage and what can be brought onboard the plane; weapons cannot be brought on board but many can be checked. The TSA website has up-to-date listings of what can and cannot be brought on planes.