The offside rules has a major impact on the way football is played by making it more difficult to play a simple offside trap or in the past by stopping "goal hanging". I was wondering what other sports have an offside rule as well and does anyone know how the offside rule in that sport affects the way it is played?
There is an offside rule in (ice) hockey, which essentially means a player cannot just hang out the attacking 1/3 of the rink and wait for the puck to be passed to him.
There is also an offside rule in rugby which seems to differ a bit with respect to the two major disciplines of the sport (union and league). The difference between offside in football and offside in rugby appears to lie in the reference point to which the position of the player is compared, in rugby that is the ball whereas in football it is the second last defender (including the goalkeeper).
While it's not called "offside", there is a similar rule in basketball, called 3-second-violation. Here you can read a nice summary of the offensive and defensive 3-second violation calls. Note that this rule exists under FIBA regulations (art. 26) as well as NBA regulations (Rule 10, sections 7 & 8).
Gridiron Football (American and Canadian) has an offside foul too. If a player is on the wrong side of the line of the scrimmage when the ball is snapped then they are offside. It is even a strategy of the offensive team to entice a player to make such a move before the snap.
You could also argue that at the kickoff, the rule that says that the kicking team have to be behind the line the ball is on is also an offside rule. A deliberate attempt of the kicking team to gain possession at the kickoff by kicking short and along the ground is even still called an 'onside kick', a play that is reminiscent of a contested kickoff or 22-meter dropout in Rugby.
Lacrosse has rules for offsides as well. The rules forces at least three players of the team to be on the attack half of the field (usually the attackers) and at least four players to be on its defensive half of the field (usually the defence and goalie).
The three players in the midfield are, however, free to move however they want.
Field hockey used to have an offside rule as well. It was very similar to the offside rule in football. However, field hockey removed the offside rule in the mid-1990s. The rule changed over time and Wikipedia is a good starting point for a historical overview and further reading on this topic.