Due to a recent change in requirements from the FAA, hobbyists are not required to register their drones. A useful chart of requirements based on activity can be found here: https://www.faa.gov/uas/getting_started/.
Although people that fly for fun are not requited to register themselves or the drone with the FAA, activity such as flying a drone for payment, research, or a club event requires the registration of the drone and a pilot in command that has a remote pilot certificate. We have a lot of resources regarding the process and study material for a license at http://uassafety.ucmerced.edu/resources/drone-license and there is a step-by-step guide on how to register a drone on the FAA site https://registermyuas.faa.gov/
Situations where you need a license:
- Photos for a club or gathering
- Any form of research
- Organized drone racing and other drone events
- Surveying for an organization
- Inspections and maintenance of buildings or equipment
- Any paid arrangement