Kids are subject to many of the same rules for flying as adults, but the identification requirements are slightly less stringent for domestic travel. Case in point, the Transportation Security Administration does not require children under the age of 18 to present identification when traveling with a companion within the United states. However, airlines may have their own requirements for identifying minors. Even if your airline doesn't require identification, a passport or birth certificate doesn't take up much space in your luggage and could come in handy if you get separated from your child.
Domestic United States Travel
If you are traveling between destinations in the United States, most airlines do not require a child under 18 to have additional identification. That said, if you already have a passport for your child, you should bring it just in case it's needed during your trip. Each airline has its own rules about minors and identification, so contact your airline a few days before your trip to make sure of exactly what you need.
That said, no matter how old they are, kids from the United States do need a valid passport for international air travel. Passport applications can take a couple of months to be processed; even an expedited application usually takes several weeks to make its way through the system. So, be sure to apply for that passport well in advance of your trip. As you pack, make a copy of the passport to take with you alongside the original, just in case. Keep each child's passport handy while in the airport, since you might have to show it at both check-in and security checkpoints.
While a birth certificate is not always required for travel, it is a good idea to carry a copy of your child's birth certificate with you. You can use it, if needed, to prove your older teen's age – some teens look older than they are and might be asked for identification. Some airlines might also require a birth certificate to prove your child's age. Because these requirements vary, always check with your airline prior to your trip to see if a birth certificate is a definite requirement, or just a smart extra.
If you are dropping off or picking up your child after they've traveled alone, you'll need to bring your own identification to complete the process. Have your child's birth certificate or passport handy as well, and be aware that security might ask older kids their name to confirm their identity.