The Q-tip basically cakes the wax onto the sides of your ear (sort of like putting cream cheese on a bagel). Over time, the wax builds up until it closes off your ear canal. It's not dangerous and removing it is very simple - they either use a tiny vacuum that sucks it out or they use a water syringe to flush it out. The wax itself is pretty disgusting, but your hearing will be pristine after they finish.
Your family doctor should be able to do it - mine did it once or twice and it went fine. But the third time, the doc had her nurse do it and she messed it up. I knew something had gone wrong when I felt pain (it should be loud and weird-feeling, but not painful) and the doctor prescribed me ear antibiotics. But it was fine - my hearing was a little muted, but it cleared up a couple of months later.