Here is a list of types of residencies and their respective lengths. This is from my understanding and may not be perfect.
New York and Delaware are the only states that currently require you to complete a residency. There are 2 basic types that are one year long:
GPR - General Practice Residency
AEGD - Advanced Education in General Dentistry
If you practice anywhere else, you can start practicing as soon as you get your license. But you may still choose to do a GPR/AEGD to further your proficiency and training.
To become a dental specialist you also have to do a residency:
Endodontics: 2 years
Orthodontics: 2 years
Pediatric Dentistry: 2 years
Dental Public Health: 2 years
Periodontics: 3 years
Prosthodontics: 3 years
Oral/Maxillofacial Surgery: 4 or 6 years (6 year includes two years of med school and awarded an M.D.)
Oral/Maxillofacial Radiology/Pathology: 3 years (i think)
There are also residencies in various disciplines of dentistry that are not considered "specialties" by the ADA, but you are still considered and expert and certified in your field:
Cosmetic Dentistry and Implantology: 1 - 2 years
TMD and Orofacial Pain: 2 - 3 years
Dental Anesthesiology: 2.5 - 3 years
Oral/Maxillofacial Prosthodontics: 1-2 year fellowship after Prost residency