I have US citizenship, my wife is a Mexican citizen working on her visa. She has a cat she can't live without and I am trying to figure out how to get it up here. Do I just talk to the airport and pay a fee?
According to the CDC, import requirements for pet cats are pretty relaxed. However, additional requirements may apply based on the destination state, county, or city.
For information about flying with a cat, contact your airline. They will probably have different rules depending on whether the cat is accompanied (considered personal baggage) or not (considered cargo). I flew my cat domestically (within New Zealand) on a 1.5 hour flight unaccompanied, so I had to pick him up at the cargo depot at the destination, rather than regular passenger baggage claim. There will probably be more paperwork required for international pet transport.
Here are the rules for taking pets to México. Note that this is a consulate in Canada, and I am Canadian. It will be slightly different coming from the United States or elsewhere (likely just veterinary association membership, but ensure you check).
When we did it, we went to the vet and got proof of health, rabies shots, and deworming - and proof of all of that! Keep ALL your paperwork, make 2 copies, keep one at home and take the originals and a copy with you.
As Greg Hewgill mentioned, ensure you've done everything the airline wants done first. In our experience, that means mentioning and paying for Pet carry when purchasing the tickets (and do this early, there is a limit on number of pets allowed on each flight).
With us, the cats travelled under the seat as our personal items. Larger pets will have to be picked up at baggage claim after immigration. Before customs, in the baggage claim area or on the way, there is a SAGARPA booth, where the agents will review the animal and your paperwork, and either approve it or require a local veterinary assessment. You will get your papers back, but they will take a copy of them for their records.
Expect this to take time, especially if there is a lineup - if there are two people travelling, have one get in line with the pets and the paperwork, and send the other one to pick up baggage and come back.
Expect the airline checkin counter (either for local onward travel, or when returning) to review all the paperwork before issuing boarding passes as well. This is why I recommend bringing two copies, in case SAGARPA takes one set, or you misplace it, or it gets used for something else accidentally.
Also: know what the requirements are in your home country for bringing pets back. In Canada, it's an on-site veterinary review, and that isn't cheap.