First, a disclaimer:
There are more than these 3 license classes, but those details are not important. These are the current license classes that someone can get today.

Technician
General
Extra

To get the higher class license, you must have already passed the lower one. (You can’t take the general exam without passing the Technician exam first)

First, each higher license gets more privileges on the airwaves. It can be summarized like this:
Technician: All VHF and UHF bands + some voice on 10M + CW on most HF bands
General: Some privileges on all HF bands + Technician privileges
Extra: All allocated amateur frequencies.

Volunteer Examiner
With the frequencies, there is also the ability to administer the exam.
General: Can administer the exam to Technicians.
Extra: Can administer the exam for any class.

CEPT/Reciprocal License
There are many exceptions to this, but in general, most countries require an extra class license to operate using your American callsign in foreign countries. There are some where General is also accepted, but it simplifies the process to have an Extra.

Callsigns
With a Technician license, you are able to request a vanity callsign that can be 1×3 (N7CTB) or 2×3 (KI7IGV).
In the past, Advanced license holders could request callsigns that began with A and were 2×2 (AA7ZZ).
Extras have the chance to request 1×2 and 2×1 callsigns. These callsigns are in high demand because shorter is easier to hear and faster to say on HF.