Puppies like to have fun, and their idea of fun is roughhousing, chewing, jumping and climbing over their friends. Don't feel like you have to engage in this type of play to have fun with your puppy -- he can learn other, safer ways to play. You're not doing him any favors by allowing him to nip and jump when he is small. When he gets older -- and bigger -- these games won't be fun and you will have to work much harder to break the already established habits.
Act like another puppy. Puppies learn how rough they can play by how their littermates act during playtime. When one puppy is too rough, the other will yip and even stop playing with him for a time. Pretty quickly the puppy learns how rough he can play.
To replicate this, make a "yip" or "ow" sound when your puppy nips you. If this doesn't stop him, pull away and stop playing with him or providing attention. Repeat the process until he learns that rough play means the game is over.
Teach him to keep his mouth closed. If he doesn't seem to catch on to your attempts to train him to be easy, gently wrap your fingers around his muzzle and hold it shut for about five seconds and say "close." Don't squeeze his muzzle, hold it just firmly enough that he cannot bite. Repeat every time he nips at you. Gradually increase the amount of time you hold his muzzle shut, never holding his mouth shut longer than 15 seconds.
Greet your puppy by getting on his level. Many times puppies jump in excitement, trying to get up where the fun is. Beat your puppy to the punch by squatting down and extending your hands when he comes toward you. You can great him with enthusiasm, pats and praise, and he learns he can keep all four feet on the ground.
Turn to the side to discourage jumping. If your puppy jumps out of excitement, even after you have greeted him down at his level, don't reward his behavior by providing attention. When he goes to jump, rotate your body to the side, so he is jumping toward your hips. Keep your arms crossed and don't pay any attention to him. When he settles down, with all four feet on the floor, stoop down and pat him.
Even when your puppy learns that he should not nip and jump at you, keep an eye on him around children, who are lower to the ground, louder and more enticing to play with than adults.