Playing the "Off-Wing"
When a right handed player is playing left wing or when a left handed player is playing right wing, they are said to be playing the off wing. When you're playing the off wing it enables you to enter the offensive zone with your stick to the middle of the ice which gives you a better angle when you take shots on goal. Also, when you're carrying the puck along the boards and a defender is trying to force you to the outside, you're able to drop your bottom hand to ward off the defender while you control your stick with your top hand and drive to the net (see the video below for a demonstration).
As an off-winger, it's important to learn the skill of catching passes on the backhand. This is especially important when you're breaking out of your defensive zone. Another advantage to playing the off wing is that you'll be able to fire off one-timers much more easily because your forehand is open to the net. Some players may find themselves more comfortable on the off wing while others prefer to play the natural wing side. It's a matter of preference based on the individual player's style of play.