@TheBreadmonkey It's like rugby, except where it isn't. Here's a handy cheat sheet:
Scrums are linear, so nobody links arms. It's called a scrimmage, and teams take other sides of the "line of scrimmage." The ball is (usually) just tossed back to the quarterback to start play. Play stops when somebody gets tackled (or scores, or the ball is out of bounds). Then they line up to essentially scrum again.
They don't practice rugby-style passing much, although it happens occasionally, so it's usually a handoff or a pass to one guy who runs like hell. A runner who passes the ball "laterals" it to a player in line with, or behind the ball holder. This makes for weirdness with the quarterback, who passes forward (a forward pass) only behind the line of scrimmage, but may lateral anywhere.
Getting the ball over a goal line is worth the most points (6). A series of plays ending in a kick through the goal posts is worth 3 points. Tackling an opponent in their end zone is worth 2 points. Crossing the goal line with the ball in a play immediately after a 6-point play is worth 2 points. Kicking the ball through the goal posts immediately after a 6-point play is worth 1 point.
A penalty is whatever the referees say it is. They get to toss a weighted flag into the air. They seem to like to do this.
There are a lot of ways a ref can call a penalty, and some teams make a lot of them, and that slows play, so sometimes refs seem to ignore a few. Common ones include offsides, which is lining up over the line of scrimmage (where the scrum happens), or encroachment (defensive player jumped over the line of scrimmage before the ball moved), hands to the back (can't shove any player in the back), pass interference (hands, bumps, tripping, lots of things), and unnecessary roughness (not proper to stomp on your opponent). Lately, you can't scissors a player with the ball by hitting him high and low. Too many injuries.
Linemen (the players actually in a scrum) have rules for where their hands can grab opponents. Sometimes, when they're all piled up, rules are wishes.
I'm not going to explain kickoffs, because the NFL, in particular, keeps changing the rules to reduce injuries. They kick off to start play after a score, or to begin the game. Then guys run. Penalties will be called, usually.
Save this for future reference. Hope it helps!