Isolate an Out-of-Control Puppy
When puppies are playing, it is natural for them to get out of control. At that time, they don't realize that their jumping and nipping has gotten wild and possibly painful. It is your job to teach them this lesson the same way their litter-mates would. If you watch puppies playing together in a litter, you will notice that if one puppy bites too hard or gets too out of control, the other puppy will yelp, and all of the puppies will turn their backs and isolate the rambunctious puppy. That teaches the puppy that no one will play with him if he is too rough.
Now that your puppy is living in your home instead of with his litter, you have to teach him to play by your rules. As soon as he bites your skin, you want to yell "ouch," even if it doesn't hurt, get up and storm out of the room, leaving your puppy behind. He will obviously be very confused at first, but if it happens every time he bites someone in your family, he will quickly learn that mouth on human skin is not allowed.
Once you have done this a few times, say "ouch" and see if your dog stops nipping. If he does, reinforce good behavior by praising him and continuing the game. If he keeps nipping, leave again.
Replace with a Toy
If your puppy tries to initiate a game with you by nipping, replace your hand with a toy and play a game. As long as your puppy is playing with the toy, the game continues. As soon as your puppy nips you instead, drop the toy and ignore the puppy. Get up and walk away if you have to. Again, be consistent. If you do this every he nips you, the nipping behavior will stop quickly.
Utilize Timeouts
If you and your family don't want to storm out of the room each time your puppy nips, make him leave the room when he has misbehaved. Keep a thin, 2-foot leash or ribbon attached to his collar when you are home with him. As soon as his play gets out-of-control and he starts nipping, say "ouch," grab the leash and lead him to a timeout. This can be in his crate or an empty room like the bathroom. Don't talk to him or look at him. Let him out after a few minutes as long as he's not barking.
Consistency is the key to teaching your puppy to stop biting. Whichever method or combination of methods you use, you must do it each time the puppy nips. If you are consistent, you will have a well-behaved companion.