The game of American Football is perhaps not quite as complicated as it may seem. Two teams compete on a filed 100 yards long to get a football from one end to the other.
The two teams field 11 players on the pitch at any one time. These 11 players are either the offense (if the team have the ball), or the defense. The offense has four goes (downs) to move the ball ten yards or else the opposition get the ball at that point and the teams switch from offense to defense.
The first time the team try to move the ball forwards (either by running or passing) they are said to be on first down, and have ten yards to go. This is known as first and ten. If they succeed in moving the ball forwards say two yards they are then on second and eight. If they succeed in moving the ball the required 10 yards in the four downs, they return to first down, and have a further four downs to move the ball a further ten yards.
It is the job of the defense to stop the opposition moving the required distance, or at the very least, stopping them from scoring.
Scoring is achieved when a member of the offense takes the ball into the end-zone, which is the very end of the pitch, not included in the 100 yard length. This is a touchdown and scores the offense 6 points. The offense are then able to score extra points either by kicking it through the posts (1 point) or running or passing it into the endzone again (2 points).
That's about it. The basics of American Football.