I easily remove about 20 ticks a year from my two dogs. When I used the old way, which was get your tweezers as close to the head of the tick and pull straight out, it would often leave parts of the tick behind. Ticks seem to like areas near the eyes and ears, which would leave marks and scars that wouldn't ever go away.
So how do you do it?
Rub the tick in a circular motion and pretend you're trying to make the tick dizzy. I generally stick with one direction, clockwise. It usually takes less than a minute, so if it's not working, make sure the body of the tick is moving around. Remember you can't get dizzy unless you're actually moving in a circle, quickly. I doubt the tick actually gets dizzy, but they sure don't like something about the movement and they crawl out on their own.
This was too difficult to document with photos, so here's the video.
Step 1: More information
This method should work on people as well, but I haven't had the chance to try it out. I don't think it will work as well, because you generally catch the tick early on and there's no body to move around. It's also easier to get parts of the tick out as you can use tweezers and then just like removing a splinter, get the rest out.
Once you have the tick out, wrap the tick in a piece of tape to kill it. If you're taking it off yourself or another person, you probably want to put it in a jar of rubbing alcohol and get it tested for Lyme disease.