They snap it to the punter, he kicks it terribly high up in the air, and it generally lands inside the other team's 20 yard line. The other team puts a player, called the punt returner, back there to receive the kick. Everyone on the receiving team tries to stop this from happening. Its quite touching really. It doesn't matter how high you kick it, as long as its still between the posts.
Punts are drop-kicked.
This is similar to the punt in that you give the other team the ball. However, kickoffs only happen after you score. Here's your stinking ball back, see if you can do the same to me. Again, the kicking team does their best Kamikaze impression, and the receiving team does their best to stay alive. It is one of the more exciting plays in the sport and happens at a surprisingly quick pace.