In the United States, the process of obtaining an amateur radio license is overseen by the Federal Communications Commission, better known as the FCC. The FCC has outsourced the process to several organizations known as Volunteer Examiner Coordinators, or VECs. To get an amateur radio license, you must pass an exam given by a VEC. You can find an upcoming exam in your area here.
There are three license classes in the US: Technician, General, and Amateur Extra. (There are also two more license classes, Novice and Advanced, but new licensees can't qualify for those classes.) Technician is the easiest test to pass, but has the least privileges; Amateur Extra has the hardest test and the most privileges.
Of course, in order to pass the exam, one must know the material. The traditional route to learn is to take a class, and/or read a license manual. Classes and manuals are offered by the Amateur Radio Relay League (ARRL), the national association for amateur radio operators, and also W5YI-VEC. You can find upcoming classes here.
Amateur radio clubs are a great resource for getting a license. Many clubs offer license classes and license exams (for a VEC such as the ARRL). You can find an ARRL-affiliated club close to you here.
The question pools for the exams are public, and so there are online practice exams, like this one. The practice exams are a great way to study for the exam.
You don't have to be a citizen, or even live in the US, in order to hold a US license; but you may not be an agent of a foreign government. You must be able to receive domestic US mail, but that territory includes FPO/APO addresses and US territories.
- You must pass element 2 to get a Technician license.
- You must take and pass elements 2 and 3 to get a General license.
- You must take and pass elements 2, 3 and 4 to get an Amateur Extra license.
You can take all three tests the same night, but most testing teams require you take an pass them in order. If you fail element 2, then you would not get a license even if you pass the other.