For future reference. Dab the tick with alcohol (some people cover the tick with vaseline), Wait a few minutes, then grab it firmly with a pair of tweezers as near to the skin as possible, then unwind the tick in an anti clockwise direction. Apparently they wind themselves into the animals skin, which is why if you try and pull them out they often leave the mouthparts behind.
I also use spot on flea and tick treatment which you put on the back of the neck/shoulders of your dog or cat. It works pretty well but doesn't stop the ticks attaching as far as I can see, it simply kills them when they do.
Good luck. I'm sure the vet will sort out the problem. But the other poster is right, you must go and get your dog checked out before the site gets infected.