To add a little more detail:
To start with, there’s a clear and distinct notion of “possession”. At any given moment, one team has possession of the ball, and the other team doesn’t, and although this can change abruptly in the middle of a play, it usually doesn’t. The distinction is strong enough that there are entirely separate teams-within-the-team for playing offense and defense: When you have possession, you have 11 offensive players on the field with all of your defensive players on the sidelines, and vice versa (plus other players for special situations, like kicking, which is not part of an ordinary play).
It also has clear and distinct plays. A play starts with the players on the two teams lined up on opposite sides of the line of scrimmage (the line showing where on the field the last play ended). One player on the offensive side (the center) snaps the ball to another player (the quarterback, generally considered the most important player on the team), and that’s what starts the action. The quarterback can then throw the ball to one of his teammates (this is most common nowadays) who then attempts to run even further with it, or he will sometimes choose to run with it himself. Although there are occasions where the receiver (the guy who caught the pass) is allowed to throw it again, this is relatively rare. What kind of play they’ll try to do is usually planned out in advance, at least roughly: You might have guys running far down the field before the throw to try to catch a long pass, or you might have them sticking close to catch a short pass before running, for instance.
The play usually ends in one of three ways: If the quarterback throws a pass, but nobody catches it, that’s an incomplete pass. The line of scrimmage doesn’t move (it doesn’t matter where the quarterback was before throwing, or where the ball ended up), and the game clock is stopped until the start of the next play. If the person carrying the ball moves out of bounds, then the game clock is stopped, and the line of scrimmage is moved to where the ball went out. If the person carrying the ball is tackled, then the game clock keeps running, and the line of scrimmage is moved to where the ball was when he went down (so you’ll sometimes see the guy getting tackled reach forward to get an extra couple of feet on the ball).
Alternately, there are a couple of ways that possession can change in the middle of a play. If you throw the ball, and someone on the other team catches it, that’s an interception. Alternately, if the person carrying the ball drops it, that’s a fumble, and anyone (from either team) can try to recover it. In either case, play continues until the new guy gets tackled or goes out of bounds: In practice, this is usually almost instant for a fumble, but can be a while for an interception. These plays can be a bit interesting to watch, because during them, everyone’s outside of their specialty: Defensive players will be trying to play offense, and vice-versa.
Alternately, of course, a play can end with a score. The most common kind of score is a touchdown, where one of your players has control of the ball in the other team’s end zone (note: It’s not enough for the ball to be there out of your control). This can either be from carrying the ball while running across the goal line, or from already being in the endzone when you catch a pass. This is worth six points, but it also gives the scoring team the opportunity to kick the ball through the goal posts from fairly close for an extra point. At the pro levels, the extra point is almost always successful, so football scores are often a multiple of seven points. Alternately, a team can, instead of attempting an extra point, can make one more play and attempt to effectively score another touchdown. This is called a two-point conversion, and as the name implies, it’s worth two points. In practice, though, since it’s less likely to succeed, teams don’t usually try to do this.
The second-most-common type of score is a field goal. In any play, a team can attempt to kick the ball through the goal posts, for three points. Since this is worth less than a touchdown, you usually only try it once it looks like the touchdown isn’t a realistic possibility. In principle, you’re allowed to try a field goal from anywhere on the field, and in moments of desperation it’s even been done from over half the field away, but in practice, you don’t usually try it unless you’re fairly close.
Finally, there’s a score called a safety. If for some reason you end the play with the football in your own end zone (for instance, the quarterback is behind the line to throw, when the other team tackles him), the other team gets two points. This is fairly rare, especially at the pro level.
Now, then, what N9IWP said about downs: You start off at “first down”, with a line ten yards ahead of the line of scrimmage that you’re trying to cross (this is abbreviated as “first and ten”). If you cross it, then you’re at first down again, and the line to cross is moved ahead accordingly. If you don’t, then the down number increases by one. So, for instance, if you manage to make it four yards before getting tackled, then it’s now second down, and you need to get six more yards (abbreviated as “second and six”).
If it’s fourth down and you still don’t manage to cross the line, then you lose possession. But this doesn’t happen very often, because most teams, on fourth down, will do a special play called a punt: In a punt, you kick the ball, and the other team recovers it and can try to immediately run it back. This always gives the other team possession, but usually moves the ball a big chunk of distance downfield, which is considered better than a significant chance of the other team getting the ball right where you are.
Most penalties are awarded by moving the line of scrimmage, in favor of the team who didn’t commit the penalty. Depending on the penalty committed, the line can move 5, 10, or 15 yards, and the down might or might not be replayed. Hence, although first down is usually “first and ten”, you can sometimes see (for instance) “first and fifteen”, if the offensive team committed a 5-yard penalty. Unlike some sports, it’s almost never worthwhile to deliberately commit a penalty.
The entire game lasts for one hour of game clock time, divided into four 15-minute quarters. Note that there are a lot of things that stop the game clock, so the actual duration of a game is usually more like three hours. At the start of the game, the start of the third quarter, and after every score, there’s a kickoff to determine where the ball will start off: The teams start at opposite ends of the field, and one team kicks the ball as far as they can, with the other team trying to recover the kick and run it back as far as they can. After every quarter, the teams switch which side is which, to even out the effects of wind, sun on one team’s eyes, etc. After the first and third quarter, the game picks up right where it left off (or rather, at the mirror image of where it left off), while at the end of the second (“Halftime”), it doesn’t matter where you were, because you’re going to kick off again for the second half. So you’ll often see high-risk, high-reward plays attempted at the end of the half, just like you would at the end of a game. Halftime usually features some other sort of entertainment, like marching bands in high school and college or (for the pros) pop singers.
That’s enough to get started; any questions?
