Pets who fly in the main cabin -- such as cats -- must have their own reservations. (It's also a chance for the airline to collect its $100, one-way fee for bringing a pet into the cabin.) American only allows up to seven pets per flight (a maximum of five in coach) on most flights, but limits that number to two pets on regional jet flights run by American Eagle.
Carry-on pets are not allowed to or from Hawaii or on trips across the oceans.
Checked pets travel in kennels in a hold under the plane, just like checked luggage, though they are placed in a special pressurized and temperature-controlled area. But that doesn't necessarily mean it is comfortable. In winter, airlines may require documentation certifying that your pet is acclimated to temperatures lower than 45 degrees.