1. Touchdown: 6 points. When a team gets the ball to the endzone they score.
2. Field Goal: 3 points. NOTE: this "field goal" is different from the extra point field goal.
3. Safety: 2 points (when the defense scores)
A team can also pick up two points by tackling an opponent possessing the ball in their own end zone. This usually occurs when the offense (the team with the ball) is very close to their own endzone (their starting point before advancing the ball down the court to score in the opponents endzone). When the ball is snapped to the Quarterback, he may have to run backwards into his endzone before passing the ball. If he is tackled while in the endzone the other team gets a safety and gets 2 points. This is the only time that the Defense can score points in football.
To summarize:
Touchdown = 6 points
Extra Point Conversion = 1 point
Two-Point Conversion = 2 points
Field Goal = 3 points
Safety= 2 points