The procedure varies a bit. In all the cases I’ve experienced, it was local anesthesia with one or more injections of anesthetic. The injections may sting a bit (both the needle and the drug), but after that there’s no pain.
I’ve experienced three different technqiues.
a metal tube with a sharpened edge is pushed into the mole, making a circular cut. The doc then grabs the mole with forceps, lifts it proud of the surrounding skin and then cuts it free. The metal tube may or may not be larger than the mole itself. If it’s big enough, you may need a stitch to close the resulting wound.
the doc uses a scalpel to shave the top layer of the mole off, leaving the skin beneath intact.
If they’re more concerned about the nature of the mole, they’ll remove the entire thing, along with the skin beneath it, plus a margin of healthy skin around it. I had one taken off of my back 11 years ago like this. They made a “football”-shaped cut (like the patches on this plywood) around the mole, and then removed the full depth of skin. Five stitches were required to pull the opposite sites of the “football” back together. They cut a football shape rather than a circular shape so the skin doesn’t end up wrinkled or puckered when the pull the opposite sides together. In my case, the biopsy of this partular mole came back as “atypical,” meaning it wasn’t yet cancerous, but it had the potential to become cancerous at a later date, so removing the whole thing along with a clean margin was the right choice.
In all cases, there wasn’t much pain after the fact. The stitches were a minor annoyance, in that the pulled uncomfortably on my skin if I stretched the area, but that was easily avoided until the stitches were removed a week later.