"When the field gets muddy in baseball, they stop the game. When the field gets muddy in football, the players roll in it like boar hogs"
-Ira Berkow, sportswriter
If a team is successful in moving the ball all the way down the field and they move the ball past the end of the field, into the "end zone", (the goal area), they score a touchdown, which is worth 6 points. A touchdown can be scored either by having a ball-carrier run into the end zone, or, by having a player catch a pass while in the end zone.
Once a touchdown has been scored the team that scores is allowed a Try (often called an "extra point conversion"). A try is usually a kicking play from 2 yards from the goal line. If the ball is kicked through the goal posts it's worth 1 point. However a team can instead attempt a 2-point Try if they want. Instead of kicking the ball through the goalposts from the two-yard line, the scoring team has to successfully move the ball past the goal line (as if it were another touchdown), this kind of Try is worth 2 points.
(Note: in the OFL teams are automatically given the 1 point try, without having to actually kick. Therefore touchdowns are worth 7 points. They can decline the automatic 1 point try and attempt a 2 point try instead. However if they fail this 2 point attempt they are not awarded the 1 point they would've had otherwise and they end up with 6 points instead of 7.)
At any time the offense may decide to kick a Field Goal. A Field Goal is when the ball is kicked off the ground by either a dropkick (where the kicker bounces the ball off the ground and then kicks it), or held by another player, and goes through the goalposts. A field goal is worth 3 points.
(Note: in the OFL the goal posts consist of two players pretending to be goal posts.)
Since a field goal is worth 3 points, (half of a touchdown) and no Try is allowed after a field goal, teams usually attempt to kick field goals only when they reckon they can't score any other way, or perhaps at the very end of game when they trail by 3 points or less and are too far away to reasonably attempt a touchdown. Most field goals are thus attempted on Fourth Down.

A field goal attempt
The only other way of scoring points in American football besides a touchdown, a try, or a field goal is called a safety and is worth two points. A safety occurs when the ball carrier is tackled in his/her own end zone, or the team with the ball purposely or accidentally drops, throws or kicks the ball and it bounces out-of-bounds from inside their own end zone. When this happens the team on defense is awarded 2 points AND gets possession of the ball. While it was originally thought of as a tactical option for the offense, it is fairly rare these days.