Each team has eleven players on the field on the time that represents either the offense or defense. While some players may be brought in for both on occasion, most players are fully dedicated to one side or the other.
The point of the game is to get the ball down the field and into the "end zone". A drive to the end zone starts when one team kicks the ball the opposing team, either as a "kickoff" or as a "punt", or a team might even turn it over. At this point, the team with the ball uses their offensive players to try to get the ball down the field while the defensive players attempt to stop them from driving down the field. The offense essentially has four plays to move ten or more yards, and failure to move it this far will switch possession to the other team, and the other team will bring their own offense out to try to move it against the other team's defense. Normally, though, a team only attempts to gain ten yards every three plays and uses the fourth play to punt it to the other team. The strategies change, though, as you are nearing the end of a half, or near the end of the game.
On the defensive side, the players attempt to stop the ball from moving. Ideally, they want to sack the quarterback or tackle the running back before he reaches the "line of scrimmage", which is how far the ball was at the start of the play. Obviously, this makes it so the offensive team has to move further in order to achieve a "first down" (which results from the aforementioned ability to move the ball ten yards). It is also possible for the defense to cause the opponents to "fumble" by dropping the ball before the player hits the ground, or they may intercept a ball that the quarterback throws. If this happens, then the defenders can actually try to score for their own team.
On the kicking side (often referred to as "special teams"), one player punts or kicks the ball to the opponent. The "kickoff" is what happens at the start of a half or after a team scores. This cannot be blocked by the team returning the kickoff. If the ball goes into the opposite "end zone" or out of bounds behind the end zone, then the team can automatically start the drive at their own 20-yard line. If it goes out of bounds down the sideline, then I believe it goes to the 40-yard line (this is very rare, though). It's also possible for a team to try an "onside kick" in which the kicker tries to position the kick so that his own team recovers it, but it must go at least ten yards before his team touches the ball (this is very hard to accomplish and is very risky). The special teams also handles "field goals" and most "extra points", which are other ways to scores. The former is the result of getting close enough to the end zone for the kicker to kick it through the "goal" (not sure of the exact term) for three points, and the extra point is used to score one extra point on top of a touchdown, with a touchdown being six points (a potential for 7 overall). Sometimes, though, a team tries to make a short, one-play drive into the end zone for two extra points, giving a total of eight. Anyways, on the field goals and extra points, the other team wants to try to block the ball, but hitting the kicker is a huge no-no. There's also the possibility a team will punt the ball, which results from failing to reach a first down and failing to get within range for the kicker to go for a field goal. Again, the receiving team wants to block the punt, though there is the possibility that the team receiving the punt can run it in for a touchdown.
Now, this doesn't even begin to touch on what goes on behind the scenes. Both sides are constantly calling plays attempting to outthink the other team, and many times, the play will change even as every lines up (Peyton Manning, former quarterback of the Colts, now playing for the Broncos is famous for doing this). The offense wants to choose if they should run or pass the ball, and passing generally comes with a play that will allow multiple options for the quarterback, the guy passing the ball. The defense generally wants to find a way to stop whatever the offense is doing, and it requires picking the right play for the right situation. Obviously, how the players execute it, both in their perception of what is going on and in the more physical ways, does have an influence how each team does. Still, football is only a partially physical sport. So much of it does come down to the strategy, but I'm saying that as someone who finds the strategies more appealing than the actual physical nature of the sport.
That's pretty much the basics. Of course, watching the sport and listening to the announcers goes a long way in figuring out the sport.
