An official (referee or linesman) is on the ice with the players, they wear a striped jersey to distinguish them from the players. Referees are mainly responsible for signaling goals, stopping play when the goalie covers the puck, when the puck goes out of play. They are also responsible for calling penalties. A linesman on the other hand, watches for Offside and Icing violations and stop play for them. They also conduct all faceoffs (except for at the start of a period and after a goal) Linesmen can also give the referee insight on a play and help to make the final call. A referee pretty much just monitors the game to make sure its as far as possible.
There are a variety of different referee systems, used in various leagues. The most popular is the 1 referee, 2 linesmen system. One referee covers the full area of the ice, while the two linesmen make offside and icing calls. In the NHL, and other pro leagues, they use 2 referees and 2 linesmen. This as helped for the crackdown on obstruction and to have stricter penalty calls. They also use the "two man system" Where both officials act as referees and linesmen.
Theres the 1-1 system. 1 referee, and 1 linesmen. The linesman covers both lines. This, and the two man system are usually used when 1 official doesn't show up.
The 2-1 system. Two referees and 1 linesmen is a newer system just being made popular in the last few years. Not many leagues use this yet.
In non competitive leagues, they use the 1 man system. This doesn't allow for as precise calls.
(some of this is from the "officials" article, on Wikipedia)
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They enforce the rules of the game - in the same way a football referee or cricket umpire enforces the rules !
They ref the game...
They call pentalies, goals, and everything else in the game.