Exchange Sacrifices
Actively sacrificing a Rook depends on you having an open file and
something to aim at. So the best-rehearsed exchange sacrifices
are ...Rc8xNc3 in the Sicilian, ...Rf8xNf3 in the French and to some
extent the Dutch, Rf1xf6 against the Sicilian... You need some
compensation for your Rook: either a good central pawn, or the
weakening of the opposing King's defences, or all three!
White sacrifices for attack on f6 in the Sicilian
White sacrifices for attack on h5 in the Sicilian
Black sacrifices for attack on c3 in the Sicilian
Black sacrifices for initiative on c3 in the Sicilian
Black sacrifices for too little on c3 in the Sicilian
Black sacrifices for initiative in the King's Indian
Encore!White sacrifices for all sorts of compensation in the Grunfeld
Another common theme is the 'passive' sacrifice of Rook for the
opponent's better Bishop by parking it on the natural outpost on an
open file. You get a passed pawn and play against a colour
complex.
Black passively sacrifices for initiative (1)
Black passively sacrifices for initiative (2)
Compensation for the exchange
Rooks don't like defending against passed pawns
The defensive sacrifice for defensive reasons is particularly associated with Petrosian, and everyone cites this example:
A defensive sacrifice
Petrosian is at it again
The Karpov sacrifice
A whole Rook