Goal of the game: move the ball into the opposing teams end zone. This equals a TOUCHDOWN (worth 6 points)
Immediately after the touchdown you automatically get the choice between an EXTRA POINT (+1 point) - kicking the ball through the goal posts from the 30 yard line
Or
A 2-POIN CONVERSION (+2 points) where the ball is placed at the 2 yard line and the offense tries to run or pass it into the end zone.
If you are trying to get to the end zone but only make it close to your goal and are out of downs or time you can settle for a FIELD GOAL (worth 3 points) attempt. It is simmiler to the Extra point except it is done from wherever you are on the field, on any down and is worth 3 points instead of 1.
Safety= when your defense tackles an opposing ball carrier in thir own end zone.
So Touchdown=6 points
Touchdown with Extra Point= 7 points (6+1)
Touchdown with 2-point conversion= 8 (6+2)
Field goal= 3 points
Safety= 2 points
That is how you score. Some of the logistics:
Your offense has 4 downs to move the ball forward 10 yards by either running or passing the ball. If you have made four plays (downs) and haven't moved the ball 10 yards the opposing teams offense takes over from where you are, which is usually not desirable. You will usually see a yellow line on the field (on TV) this is the line that marks the 10 yards you need to gain. Once you make it past that line, wherever you are downed they measure another ten yards and you are given 4 more downs (plays) to gain 10 yards. You will hear it referred to as "getting the first" or "first down" or "moving the chains" (referring to the chains they use to measure the 10 yards) and often you will hear "first and 10" or "3rd and 5" that is what down and how may yards left to get a first down.
Usually if you don't get a first down by the end of the third down you use your fourth down to PUNT the ball. A kicker comes in behind the line, gets the ball snapped to him and "drop kicks" the ball. The point of this is to push the ball as far back and away from your end zone before the other teams offense takes over so that they have farther to move the ball before they can score. This puts a strong emphasis on third down because on fourth down is usually used to punt the ball to the other team. In certain situations the offense might choose to "go for it" on 4th down. Usually only in desperate situations with short distance to go for a first.
Kickoff- A team kicks across the field from a ball teed on the ground to the other team. The goal is to kick it as far back without kicking it into or past the end zone that allows your team to tackle them as far back as possible. If the ball goes into the end zone and the receiving team takes a knee or, if the ball goes passed the end zone it is a touchback. A touchback takes the ball automatically out to the 20 yard line. A kickoff takes place at the start of the first and third quarters. Who initially kicks the ball is decided by the winner of the coin toss. A kickoff also takes place after a score (The scoring team kicks to the other team)
Sorry for shitty formatting. On mobile, I also got lazy toward the end.