Off-Sides in its most basic definition states, the puck has to cross the line into the offensive zone before anyone on the attacking team.  The puck is the key and the blue line is the door, you have to use the key to unlock the door.

In the diagram above, F1 has the puck, and is on the attack.  F2 crosses the blue line ahead of the puck with both skates.

If a player has BOTH of their skates across the blue line, without possession of the puck, the play is called off-side, and a face-off will be in the neutral zone.

Delayed Off-Sides:  In hockey, a delayed offside occurs when an attacking player enters the offensive zone BEFORE the puck, but they do not touch the puck or gain possession of it.  The play is not immediately stopped, and the attacking team must clear their zone by crossing the blue line before anyone can play the puck again. 

If a defensive player brings the puck into their own, defensive end, and an offensive player is still in their (defensive team's) zone, the offensive player can play the puck and is not considered off-side.