Yes, the FAA has a whole list of licences that cover aviation including hot air ballons. In FAA parlance you need a "rating" for that particular air craft.
To earn your lighter than air pilot certificate, you need to attend a ground school or do a home study course to get the basics necessary to pass a FAA written exam. The FAA requires at least 6 hours of flight training with an instructor, 2 solo flights, and 1 altitude flight. The final step is to pass an oral exam and a flight exam with a FAA pilot examiner. Passing these steps will earn you your private pilot certificate for hot air balloons.