What are the rules of hockey?

there are two teams playing against each other with 19 players on each side. Generally 12 forwards, 6 defenseman, and 1 goalie. Sometimes a team will choose to play 7 defenseman and have 11 forwards play (this is because the 4th line doesn't get much ice time anyways). 

The scoring rules aren't too bad. The team with the most goals wins. IF a game is tied after the 3 periods (or 60 minutes) then a 5 minute Overtime period will ensue. IF the game is still tied, then a shootout will decide the winner. Shootouts may take forever. But the coaches write down their top 3 list of shooters they want to go and give it to the referee before the SO begins. Think of penalty shots deciding soccer game, same deal. The shooter goes one-on-one with the goalie. The more European you are, the greater likelihood you try to do something fancy. The team with one more SO goal wins.

Teams start out a hockey game playing 5 on 5. So 3 forwards, and 2 defenseman for each team. The goalie's are also in net, obviously.

Now, how can one team have more players at one time on the ice than the other team? Answer: Penalties.

There's a fairly long list of penalties or things a hockey player CANNOT do or he will go to the penalty box for 2 minutes while the other team has a "Power Play."
The other team plays 5 on 4. There are times when yet another player can be penalized to make it a 5 on 3. Which should be a guaranteed goal for the team with 5 players. Sometimes penalties can be 4 minutes long or 10 minutes.
4 minute penalties come from high-sticking (if a player gets his stick into the face of another and causes blood to leak) or they come from boarding. 10 minute penalties come from Game Misconduct's (someone hitting another player from behind very hard or elbow the head).

Types of penalties: interference, hooking, holding the stick, high sticking, tripping, boarding, charging, roughing, goalie interference, kneeing. They're all pretty self-explanatory and you can look them up if need be. Hits to the head is what the NHL is trying to clamp down on. Any elbow to the head or shoulder to the head will lead to a review by the NHL (namely Brendan Shanahan). He will decide if a player should be suspended or not. Brendan started doing this just this year.

There are things that happen within the game that don't get penalized. But faceoffs are a HUGE part of hockey. Opening faceoffs take place in center ice. But when a team has a power play, they start out in the opposing teams end. The referee drops the puck on the faceoff dot and each player has an equal chance of winning puck possession. 

When a player carrying the puck into the opposing zone (past the opponents blue line) has a teammate who crosses the blue line a step ahead of the puck carrier, that is ruled as Offsides (just like in the NFL) and play is stopped. Another faceoff takes place. When a player shoots the puck down from his end to the other end (without crossing the red line, the halfway point in the rink) and no one touches the puck on his team, that is ruled Icing. Which means the opposing team has a faceoff in the other teams zone (the team who shot it down the ice). When a puck goes out of the rink and into the stands that also stops play (stops the clock) and another faceoff must take place.

When a player shoots the puck out of the rink from his own zone without touching the glass, that is a penalty: delay of game.

Like baseball, hockey is a game of matchups. Each team plays four lines of forwards and 3 pairs of defenseman. The first 2 lines are considered "scoring lines," as in, they're supposed to produce offense and score some goals. The 3rd line is often called the "checking line." This line is used to matchup against the opposing teams top line. The checking line's objective is to shutdown the big guns on the other team. Some good "shutdown" centers in the NHL are Jordan Staal for the Pittsburgh Penguins and Dave Bolland of the Chicago Blackhawks. These guys are a pain to play against because they're pesky and physical. The 3 forward positions are LW (left wing), C (center), and RW (right wing). The center takes all of the faceoffs when its his lines turn to go on the ice (unless he gets kicked out by the referee). The center is also often the best "passer" on a given line. When its Sidney Crosby (Penguins) centering your line, it may be the best passer AND scorer. 

And the 4th line generally plays 6-9 minutes a game, which isn't much at all.
These guys are again physical players who like to crash and bang. These are usually bigger and tougher players who often engage in fights with the opposing teams "agitator" or "instigator." Fights take place in hockey for momentum reasons. One team wants to get some momentum on his teams side to pump them up, so he'll ask the other teams player to engage with him. Fights also happen if some player hits a top player from an opposing team. Matt Cooke (Pit) took liberties on Marc Savard (Boston) a couple years ago, than had to answer the bell when Shawn Thornton (Boston enforcer) wanted to fight him in the next game, to "settle the score" so to speak. 3rd and 4th line guys are known as "role players" too because they do the little things to help win games. Penalty killing (when your team has 4 or 3 on ice while opponent has 5), or faceoffs, or shutting down the opposing teams top dogs.

At the end of a game, one team gets 2 points (a win), 0 points (a loss), or 1 point (an OT loss). Shootouts are designed to give a team that extra point. 
When a team wins 2-0, that is called a "Shutout." 

Sorry for the elongation.