http://www.chess.com/forum/view/chess-openings/difference-between-defensevariationcontinuationetc
A "variation" is usually a formal name of a common or at least well-studied series of opening moves, not really an alternative plan, although usually different variations stem from a single move as alternative moves. I'm not sure about "line": to me that is a more general term that considers *any* sequence of moves, even if played only in one game, or even if only considered during one game, without regard to any history or analysis of that sequence of moves.
I notice that there are many different variations to chess openings. When I'm watching videos on openings, I hear the word "lines" and "variations" often. Do variations mean that it's what you do if your opponent doesn't follow a certain "line" of an opening, they are like a plan b or back up plan to an opening? Thank you.