How old does a puppy have to be to fly on an airplane?

That depends on the requirements of the individual airline. Some will only accept dogs of a certain age or size. Some will allow them in the cabin or only in cargo. Some will not accept them at all.

In the U.S., nearly all airlines that do permit dogs require a current (no more than 10 days old) health certificate and vaccination certificate for rabies.

For international travel there may be regulations from the country of origin or more likely from the country of destination about the age, veterinary records and health status of a dog being imported.