An offensive rush occurs when players from a team enter the attacking zone and their number of attacking players outnumbers the defending players already in the zone. For example, if the attacking team enters the zone with three players, but only one defenseman is in the zone, it is considered an odd man rush. Some people specify the various odd man rushes by stating the number of players. In the previous example of three attackers and a single defenseman, it would be called a "three-on-one" rush.
An odd man rush gives the attacking team an advantage since they can maneuver around the fewer number of defenders and optimize where they take their shot on the goaltender. In odd man rush situations, the goaltender usually prefers the defenseman stay between the puck carrier and one of the other attackers, in hopes they can eliminate one potential pass from happening, and allow the goaltender to have a full view of the puck carrier.