Easy tutorial and basic rules when you start watching:
1. A team gets for opportunities (4 downs or plays) to gain 10 yards. Usually, when a team in on 4th down (after 3 attemps) the team opts to kick (punt) which gains territory, rather than turning the ball over at that exact spot
2. When the ball hits the groud, or the ball carrier hits the ground with both knees, the play is over. Same goes for when the ball is in touch
3. Getting to the end-zone, results in a touchdown. Unlike the word might suggest, you don't have to necessarily touch the ball down, like in rugby, where we call it "try". I love how these names are the exact opposite of what they actually mean. A TD is 6 points. After that a team gets the chance to go for a PAT (Point after Conversion) or, if they prefer, try a 2-point conversion. The team gets the ball at the 2 yard line and has 1 play to get another touchdown, worth 2 points. On 4th down, a team can try a field-goal (3 points) but teams only try this from 55 or 60 yards out tops.
4. The team is only allowed to throw the ball forward once per play. Usually by the Quarterback. It is allowed to throw the ball backwards as many times as you'd like, but it doesn't happen a lot.
5. If a player drops the ball after catching, it's a fumble, and the opposing team can pick it up and gain possession. It's very dangerous and players prefer to end the play by either going to ground or walk into touch rather than gaining that extra 2 yards with risking a fumble.