In many states you get a temporary plate from the dealer that is good for 30 days or so, and receive your permanent (or annual) plate from the state in the mail.

If you buy a used car from a private seller you have to go to the motor vehicle bureau and transfer the title and register the car yourself. If you buy from a dealer they handle that for you (and charge you a few hundred dollars as a "document processing fee").

In some cases you can transfer the plate from your old car to your new car, but that's not always possible. There doesn't seem to be a lot of rhyme or reason to it.