The lines and what they forbid
The rink is divided by two blue lines into three zones. The attacking team may not enter the opponent's zone before the puck does — that is offside, and it exists to stop players waiting near the goal for a long pass. The consequence is that attacks have to be carried or passed in with control, which is why the moment of entry is such a contested part of the game.
Icing prevents the opposite abuse. A team cannot simply shoot the puck the length of the ice to relieve pressure: if it crosses both the centre line and the opponent's goal line untouched, play returns to where it came from and the offending team may not change its tired players. That combination — a face-off deep in your own zone with exhausted skaters — is a serious punishment for a moment of panic.
These two rules together explain most of what a viewer sees. Teams set up at the blue line, dump the puck in when they cannot carry it, and fight for possession in corners, all because of where the lines are.
Penalties and the power play
Most infractions are punished by sending a player to the penalty box for two minutes, leaving his team short-handed. No other major team sport routinely removes a player from a game in progress, and the effect is dramatic: the attacking team has a numerical advantage and a much higher chance of scoring.
A minor penalty ends early if the team on the power play scores. More serious offences carry longer terms that run their full length regardless, and the most serious can remove a player for the rest of the game.
Special teams — the power play and the penalty kill — are practised as separate disciplines with their own personnel. A team that is excellent at both can win a season's worth of tight games on that basis alone, which is why the special-teams percentages are among the first numbers a coach looks at.
Overtime and the shootout
A tied game goes to a short overtime played with fewer skaters on each side. Removing bodies opens enormous space on a surface designed for full teams, and the result is a period that is close to a series of breakaways.
If nobody scores, the game is decided by a shootout: alternating one-on-one attempts against the goaltender. It is a genuine skill contest and also an artificial one, which is why it is used in the regular season and abandoned entirely in the playoffs.
Playoff games instead continue with full-strength overtime periods until someone scores, however long that takes. Those games are the sport at its most exhausting and are the reason the playoffs are treated as a separate physical undertaking rather than an extension of the season.