You do an externship while you're a dental student, usually in your 3rd and 4th years. Externships last anywhere between 1 week and 4(sometimes even more). Externships give you an opportunity to see different programs and see what residency life is like. They are for the most part shadowing/assisting roles.
Internship can have 2 meanings. First year residents are called interns. They are admitted into the residency program and are at the bottom of the chain. The second type of intern is known as a non-categorical intern. They are NOT admitted into the residency and are serving a 1 year contract and at most programs act as a first year resident(the other kind of intern). Non-categorical interns often times did not match and are trying to beef up their application for the next cycle.