Crossing the line is a cinematographic error which sometimes crops up when filming a conversation between two people, with the camera giving a third person point of view.
Imagine you are filming the conversation and are sat, say, quite close to one party in the conversation. Now, draw an imaginary line between the two actors involved. This is 'the line', (sometimes referred to as the action axis). Now, the two actors in your scene are facing eachother and you are filming the first actor. Imagine following the conversation by moving your head, back and forth, between the two subjects. You would consistently see more of the left side of one actor's face and more of the right side of the others.
OK - remember the imaginary line between the two? If you 'cross the line' for the next set of lines of one of the actors and film from here, you will be filming their 'wrong' half. This gives the viewer the impression that the camera, as observer, is moving around. This is quite disconcerting and is known as 'crossing the line'.