Do you take a free piece ( B or N ) or win the exchange?

Is there a general rule that says what you should do in this situation?

There is pretty much no "general" rule in chess, except that a checkmate is to be avoided (and since chess is deterministic, your very first move might be a complete blunder). For every other rule, there are exceptions.

However, in most positions, going for the free minor piece will be the better choice. Think of it as deterministic as it gets: if you trade all pieces in equivalent, you will either end up in a R vs (B or N) endgame if you go for the exchange - this can be quite tough to win in some instances.
If you go for a free piece, you will be able to play King + Pawns vs King + Pawns + Knight / Bishop which is a piece of cake to win even if the other side is up one, two or mostly even three pawns (of course your pieces and pawns need to be coordinated in a good manner).

To sum it up: it's all about disallowing your opponent any kind of coordination and counterplay. If you take away an entire piece, that will damage this cruicial ability way more rather than restricting them to a piece that is less favorable considering individual movement. Bishops can sometimes build very annoying fortresses!

Each situation is different. I think it's generally better accepted that a full piece is better. Especially in the middle game, minor pieces are a lot more active than rooks.

If you going to win the exchange and your opponent is left with the 2 bishops you have to seriously consider if its worth it

Yep, the "general rule" in this case is arithmetic. :)

depends on the situation. if closed position take N and if open take B

Thanks for all your thoughts on this subject. I have always found it difficult to decide what move to play in this situation. You all have been very helpful.

@jfm22 said in #1:

Is there a general rule that says what you should do in this situation?

Piece is better all things being equal.