The delay of game penalty is simply what it says. It's a penalty called when a team or player does something to deliberately delay the game, such shooting the puck over the glass out of your defensive zone; freezing, falling on, or holding the puck unnecessarily, etc.
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On every play, as soon as the officials mark the ball ready for play, a 25-second countdown begins. This is known as the "play clock." If the offensive team does not snap the ball before the 25-second clock expires, they are penalized 5 yards for "delay of game."
They are delayed penalties because the team that the penalty has been given to have not touched the puck. As soon as they touch the puck the referee will call the penalty.