On Southwest, the first two checked bags are free, up to 50 pounds. If the bag weighs between 50 pounds and 100 pounds, you will pay a $50 fee for the 3rd through 9th piece of luggage. After the 9th piece, checked luggage costs $110 a piece. Pieces over 100 pounds or longer than 80 inches are sent via Air Cargo, though you must make this arrangement ahead of time.
Transportation Security Administration rules dictate that each person be allowed one personal item, like a laptop with case or purse, as well as one carry-on bag. The TSA has a handy list of allowable carry-on items at their "Prohibited Items" webpage. Carry-on items cannot be more than 10 inches by 16 inches by 24 inches. These size requirements tend to be more lax when the flight is not full, since there is more room for each passenger in the overhead compartments. If your carry-on is oversized, you may be asked to check it at the plane door. You will reclaim it on the jetway when you get off the plane.
You have a little more leeway for carry-ons when traveling with children. Southwest Airlines allows you to add one stroller and one car seat per child, on top of your regular carry-on baggage. This applies both for children with their own paid seat and those sitting with an adult. You can check the items at the gate when you board your plane, and you will get them back when you deplane.
Assistive devices like wheelchairs are not counted against the checked baggage count. Customers can also check those items at the gate.