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

Chess Quotes

"Flip-Coin Chess: Does not pay attention to all (or sometimes even any!)
of the threats generated by the opponent's previous move.

Hope Chess: Does pay attention to all the threats generated by the
opponent's previous move, but, before making their current move, does
not check to make sure that all checks, captures, and threats by the
opponent on the next move (in reply to that move) can be safely met.

Real Chess: Not only deals with opponent's threats from the previous
move but, before making their move, also makes sure that the opponent

— Dan HEISMAN