Let's say I'm playing a game against an opponent. It's my turn, and I can get a Rook, which is not hanging, or I can get a Bishop that is hanging. What piece should I get, and why?
Depends. If you can take the rook with a pawn, that is probably better. If your capturing piece is a knight or bishop, take the hanging bishop in most cases; bishop is worth 3 pawns and winning the exchange bishop/knight for a rook is only a 2 pawn advantage.
There may be positional considerations that can change that calculation as well, so that can change which way to go.
Let's say I'm playing a game against an opponent. It's my turn, and I can get a Rook, which is not hanging, or I can get a Bishop that is hanging. What piece should I get, and why?