Usually the Orthodontist will send you to a general dentist (DDS or DMD) unless you are having impacted wisdom teeth taken out, then you will be going to an oral surgeon.
If you are awake for the procedure you won't feel any pain, but you will feel pressure during the extraction. Modern Novocain is a remarkable drug.
Here is the procedure:
First you are placed in the chair and a topical anesthetic is rubbed onto your gums at the injection site.
After it takes hold, you get several injections of Novocain, and are left alone for a few minutes until that kicks in.
Once you are good and numb, the dentist uses what is basically a pair of pliers to twist the tooth in its socket, freeing it from the jaw. Pop, it's done.
Then you get a folded gauze to place in the hole, and may get a stitch to facilitate healing.
Darvocet or some similar pain killer is proscribed, take one before the Novocain wares off.
Replace the gauze as it becomes soaked. You may have to do this several times.
It'll hurt a bit, but not much. The mouth is one of the fastest places your body heals.