It's ideal when traveling and especially if you are traveling with your pet or if your pet is traveling alone is to have no plane changes, no layovers. Sometimes this is unavoidable.
There are a few airports that are said to be particularly pet-friendly. Please add to the list if you know of any more. Please see Pet Relief Areas above for information provided by airlines, etc.
Questions to ask before the trip:
Check with your airline and the airport to find out if the pet areas you've heard of are still in operation. Does your airport or the one you're visiting have such an area? Are they in service 24-hours if that is of concern for your scheduled flight. Are they wheelchair accessible?
Check with your carrier to see how your pet will be handled during time on the ground. Will he or she be checked? If there's a change of planes, ask how your pet will be handled and how it's assured that the pet will make your connecting flight so you'll be reassured.
What happens in the event of a flight delay? Is some sort of tracking system used so your pet and his or her carrier may be located at all times?
If your pet / service animal is traveling with you, are you prepared with some sort of bowl (maybe collapsible), a means of providing him or her with water and food? A plan for a bathroom break?
When traveling with a cat, a layover may mean a walk on a leash in a pet relief area. It may also mean a small amount of litter in a pan in the carrier and any necessary cleaning up before boarding the next flight.
Use common sense and check with your vet on the appropriate way to keep your pet safe and comfortable.
A few Examples: (Many other airports have such areas)
Atlanta International Airport (ATL) has its Poochie Park.
San Francisco Int'l (SFO) boasts two pet travel areas that are available 24 hours.
Philadelphia International Airport (PHL) is said to have seven fenced-in areas for pets.
Phoenix Sky Harbor International Airport in Phoenix, AZ has two dog exercise areas.
The Bone Yard is at Terminal Four. It's said to be 2,000 square feet. The Grassy Paw Pad can be found near Terminal Three.
Denver International Airport (DIA) has an enclosed exercise area for pets. This area is wheelchair accessible.
A valid concern: when traveling with pets internationally, what do you need for layovers in different countries?
Paw-fect Airports: The Most Pet Friendly Airports In The U.S. | Layover Guide
Indulge in Layover Heaven - The Early Show - CBS
Pet Travel: Layover and Transit Requirements at Airports
[Image from public domain pictures]