The team that is on offense has 4 chances to move the ball up the field ten yards. Each play is called a "Down". So the first play is First down, Second play is second down, etc.
Each time they move up 10 yards, they get a new set of downs, so it restarts at 1st down. If they fail to move up 10 yards, then they must punt the ball. Punting is usually done on 4th down, but the team may choose to go for it. If they fail to get past the first down marker on 4th down, it results in a turnover, meaning the defensive team gets the ball where the offensive team left off.
Each time the offensive team reaches the endzone (the big empty areas on each side of the field, where the big yellow things are), they get 6points (this is called a Touchdown). Then, they get the ball 5 yards from the endzone. They have the choice to either: 1.Kick the ball in between the post, and get 1 extra point (this is called the extra point, or PAT) or they may choose to run or pass the ball into the endzone for 2 points, (this is called a 2 point conversion).
If a team hasnt reached the endzone, but are in range to kick the ball in between the post, they may kick it. They have 1 chance, and if they dont make it, it results in a turnover. Lets say Team A is in range, and its 3rd down. They kick and miss. Even though its 3rd down, the other team gets the ball.
If a team has the ball alllllll the wayyyy down the field, at the OPPOSITE endzone (the one they ARE NOT going for), and a ball carrier gets tackled in the endzone he is coming from, the other team gets 2 points. This is called a saftey. (Not to be confused with the position.)

There are many variations of the way the game is played. There is NFL rules, NCAA rules, Streetball, Arena Football, and many others. If there are any questions or corrections let me know. I will be GLAD to fix them.
(stick requested! for people who dont know about this fantastic game!)