Amateur radio (ham radio) is a licensed radio service that lets ordinary people operate their own transmitters on reserved slices of the spectrum. It exists for self-training, experimentation and public service. The word "amateur" means unpaid: you can't be paid to operate, and you can't use it for business.
What it's for
The five purposes of the service, paraphrasing the FCC's “Basis and Purpose” rule (47 CFR 97.1).
These aren't just words. They explain why the rules look the way they do: operators are trusted with real power and real spectrum, in return for staying noncommercial and keeping their skills sharp.
What you can do with it
Almost anything that involves radio:
talk locally through repeaters, or across the world on HF
send Morse code, text and pictures, and run digital modes from a laptop
bounce signals off the Moon, through satellites, or off meteor trails
build your own antennas and radios
provide communications when normal systems fail
compete in contests, chase distant stations, or operate from a mountaintop
no broadcasting to the general public, and no business use
no payment for operating, and no music
no secret codes: messages must be in a language others can read, and stations must identify themselves
How you join
You pass an exam, administered by volunteer examiners, for the class of license you want, and the FCC issues a call sign. See License classes and privileges for what each class allows, and Clubs, Elmers and community for finding help.