Watching puppies coming into the world is an awesome experience. Seeing the whole process of new life being born will give you a whole new appreciation for your pitbull and her pups. Knowing some basics is an absolute must for delivering your pitbull puppies.
The most important thing you have to establish before you attempt to deliver your pitbull's pups is trust. The female must totally respect you; not be frightened of you; accept your hands all over her body; and must not show any aggression toward you in any fashion. While this is true for any breed of dog, it is especially true in aggressive breeds, such as the pitbull. So take the time, long before your dog whelps, to get her comfortable with you and the place where she will be in labor.
Prepare for the labor with lots of dry, clean towels; alcohol; round-nosed sterile scissors; and a suction bulb. You need to have your vet's phone number handy, and a pen and paper to keep track of times between births. Make sure your dog is comfortable in her whelping box, and try and stay out of her way. The best way to help her is to let her do it herself, with minimal interference by you. You should only step in if necessary.
When you notice the first pup being born, watch to make sure that the female licks it within a few seconds of delivery, removing the birth sac. If she does not, you will need to clear the sac from the pups head, but only after the pup is out. Suction the mouth and nose, and use a clean towel to rub it briskly on its back and head. This stimulates breathing and helps clear mucus. If the mother does not chew loose the umbilical cord, you can snip it with your sterilized scissors about an inch or so from the belly. Use a touch of alcohol on it to make sure it is clean. Once the baby has been cleaned and dried, place it at the mother's teats and help it latch on.
Your main job is to watch the mother for complications. If she appears to be straining and nothing is happening for over 20 minutes, you need to call your vet. If a pup becomes "stuck," you can reach down and grasp it by the shoulders and gently pull down when the mother pushes. Only pull when she pushes or you could injure her. If mom does not clean off her babies, you need to do that and get their breathing going with lots of rubbing and toweling. Mainly, stay out of the way. A dog's natural bond with her pups can be interrupted with too much human interference. Once all puppies are delivered and nursing (they need to nurse within a few hours at most), schedule all of them for a doctor's visit, just to make sure all is well.