There are limits to how many pets can fly in the cabin of a plane on any given flight, so it's important to plan ahead and research the airline you're using beforehand. Not every airline will be the same when it comes to pets, and different companies may have different rules.
If your pet is on the bigger side, United has a program called PetSafe that specializes in flying pets who are too big to fly in the cabin with their owners. All pets must be at least 8 weeks old in order to fly. United also allows rabbits and small birds (yes, really) to fly in the cabin.
Delta requires all pets be at least 10 weeks old in order to fly. Delta allows pets to fly in the cargo hold, but that can't be booked until 14 days before your flight. Household birds are also allowed in the cabin.
JetBlue does not fly pets in the cargo hold, but it does have a program called JetPaws, which offers everything you need to fly with your pet.
Southwest does not allow pets on any international flights. All pets must be checked in at the airport ticket counter prior to their flight, and they must be at least 8 weeks old in order to fly. Southwest does not fly pets in the cargo hold under the plane.
American Airlines requires that all pets be at least 8 weeks old in order to fly. Pets are allowed to fly in-cabin to most destinations, but certain destinations have special restrictions, which are listed on American's website.
You will have to pay a fee to have your pet travel with you, and those range from $95 to $125 each way, sometimes more (for in-cabin travel). Pets are never allowed to travel with unaccompanied minors. Your pet needs to be small enough to fit comfortably in a carrier that fits under the seat in front of you.
Approved service animals are, of course, exceptions to every rule.