What are factors to consider when and when not to sacrifice? For example, imagine in a game where your opponent puts his bishop in direct sight of yours and you're not sure whether to take it or not? Is it safe to assume that one of the factors would be if your bishop is bad, then it might be a good idea to sacrifice it especially if the opponent has a good bishop?
What you describe would not be a sacrifice; it would be an exchange - bishop for bishop. And yes, if your bishop is bad (or generally unlikely to enter play without some time-consuming work), then it is a good idea to exchange, especially if your opponent's bishop is well-situated.
To address sacrificing - you should only sacrifice material if you gain a significant positional advantage by doing so, as in a well-played Greek gift sacrifice. Wikipedia's example fits here nicely:
1.e4 e6 2.d4 d5 3.Nc3 Nf6 4.e5 Nfd7 5.Nf3 Bb4 6.Bd3 O-O:
Now White plays the sacrifice:
7.Bxh7+! After taking the Bishop (
7... Kxh7) Black is two points ahead, but severely exposed. White takes advantage of this by
8...Kh8 9.Qh5+ Kg8 10.Qh7#, which Black cannot avoid without losing his queen.
For another quick read on the topic of sacrificing, I would suggest The Art of the Sacrifice, written by an amateur only a couple months ago. It goes through the main points of sacrifices, along with profuse examples. Also, the ubiquitous Wikipedia is not to be missed.
Naturally, though, good chess reading costs money, so if you want some good, solid, on-your own instruction, you could consider purchasing some books on the subject.
Sacrifices have the following goals:
- exposing the king: for example after your opponent castled, sacrificing the bishop to take the pawn diagonally to the king.
- cut the escape of the king: this one is hard to explain, but basically after the sacrifice, if the opponent recaptures, the king can no longer occupy this square. Note that this can also happen even if there was an enemy piece originally, and you sacrificed by capturing it, but under the scenario that the new piece there have different mobility (e.g usually a knight) so after a check this new piece can't block it as opposed to the old piece.
- gaining (or making the opponent lose) tempos: sometimes you find combinations that allows you to make important moves in time, or make the opponent lose time that he doesn't have. One example is if the two players are about to promote, but you sacrifice a piece so that the king blocks his own pawn or moves into a square that will be attacked by the promoted Queen (so when you promote you deliver a check).
- gain control of the center (or any other particular area): this usually happens by sacrificing pawns from the flanks to be recaptured with enemy pawns from the center.
- weaken the pawn structure: in very closed positions, these kind of sacrifices are almost always played. I can't give examples since I have never done one and usually don't understand them without computer analysis.
- removing of the defender (deflecting): when a piece is the only one stopping the inevitably, it is usually worth taking it down no matter if you lose material.
- luring a piece to unwanted squares: luring the king into forks, screws, etc. Another example is when a enemy bishop is preventing your pawns to advance, making him move out of that diagonal is deadly.
Some quick examples from one of my games:
[FEN ""] 1. e4 e5 2. Bc4 d6 3. Nf3 h6 4. O-O Nf6 5. Nc3 Be7 6. d4 exd4 7. Qxd4 O-O 8. Be3 Nc6 9. Qd3 Bg4 10. Bxh6 gxh6 11. e5 Nxe5 12. Nxe5 dxe5 13. Qg6+ Kh8 14. Qxh6+ Kg8 15. Ne4 Be6 16. Rfe1 Bxc4 17. Nxf6+ Bxf6 18. Re3 Bg5 19. Rg3 f6 20. Qg6+ Kh8 21. Rh3+ 1-0
10.Bxh6 might not be the best move from that position, but it is not a bad one either, there is a very dangerous pin by the white bishop that together with the Queen can do amazing things. The main idea is to expose the enemy king.
(#3 and #6) -
11.e5 if the Knight moved, I would have achieved #6 over the bishop, however he didn't and decided to take the pawn, I didn't care which way he take the pawn, my #3 is achieved since all I wanted to free the Queen diagonal.
15.Ne4 I attempted to remove the defender of the bishop, with the plan of capturing the bishop with this line
15.Ne4 Nxe4 16.Qg6+
16.Rfe1 is also interesting, some people would say it is not a direct sacrifice though, but it is deliberately not protecting a piece. The main idea is to don't waste any time and activate the rook ASAP, while still having a chance to protect the other piece if he decides to go for NxN first.
A good time to "sacrifice" is when the true value of the piece you are getting is worth more than the value of the piece your are giving up, even though your piece has a greater nominal value.
Examples: 1) You (White) have a rook on an open f-file, and your opponent has a knight on f6. You play
RxN, Black retakes with his pawn at g7:
...gxf6. Two things have happened. Black has lost a major defensive piece on the Kingside, and lost pawn protection on the g file. Even though you have "lost the exchange," what you have gotten is well worth a rook. 2) In another scenario, you might move
Nf5 to coax
...gxf5 from the pawn on g6. You recapture, having traded a knight for a valuable pawn that no longer protects the enemy king.
Generally, you sacrifice in an area, usually the kingside, where you have preponderance of force. If you have four pieces to one, or five to two, you sacrifice one for one or two for two, even if your pieces have greater nominal value (sometimes a rook or queen for a minor piece). Then your remaining pieces crush the enemy king.
"Material" is but one of the types of advantages in chess. However, it is the easiest one for most people to understand.
Other types of advantages: space, piece activity, king safety.
You learn how to spot good sacrifices when you become more aware of the other forms of advantage in chess. Sacrificing is merely taking a hit on the material score in order to gain an advantage in another area.
These words will be easily forgotten without examples. The best examples are classic games. [The games of modern players are not as good because their defensive technique is so good that sacrifices rarely work against them.]
Don't wait for people to teach you in this thread - start learning with a google search or two. Here's one on Great sacrifices in chess.
Go over the many games there and learn from them. See if you can get a sense as to what compensatory advantages are gained for the sacrificed material.
Most of the examples presented in the other answers are about sacrifices related to tactics, i.e. you give up material for a clear purpose; to mate the enemy king or to gain back the material (and more) soon. While these are certainly valid sacrifices they don't happen all that often in high level chess, because everybody is doing their best to not allow them.
Much more common at those levels are positional sacrifices, where you give up material in order to gain a long-term positional advantage or even just to unbalance the position. Most common are pawn and exchange (you give a rook for a bishop or knight) sacrifices.
Typically exchange sacrifices happen in rather closed positions where the rooks are not that powerful and often the captured knight/bishop was a very valuable piece (for defense or attack).
Pawn sacrifices are often done to spoil the opponent's pawn structure or coordination of pieces.
As an example this very nice positional queen sacrifice by Caruana. Look at the position around move 21. Black is completely dominated on the light squares. While it is not clear how black will lose it is a pretty hopeless position.
[Event "London Chess Classic"] [Site "London ENG"] [Date "2016.12.15"] [EventDate "2016.12.09"] [Round "6.3"] [Result "1-0"] [White "Fabiano Caruana"] [Black "Hikaru Nakamura"] [ECO "B96"] [WhiteElo "2823"] [BlackElo "2779"] [PlyCount "63"] [FEN ""] 1. e4 c5 2. Nf3 d6 3. d4 cxd4 4. Nxd4 Nf6 5. Nc3 a6 6. Bg5 e6 7. f4 h6 8. Bh4 Qb6 9. a3 Be7 10. Bf2 Qc7 11. Qf3 Nbd7 12. O-O-O b5 13. g4 g5 14. h4 gxf4 15. Be2 b4 16. axb4 Ne5 17. Qxf4 Nexg4 18. Bxg4 e5 19. Qxf6 Bxf6 20. Nd5 Qd8 21. Nf5 Rb8 22. Nxf6+ Qxf6 23. Rxd6 Be6 24. Rhd1 O-O 25. h5 Qg5+ 26. Be3 Qf6 27. Nxh6+ Kh8 28. Bf5 Qe7 29. b5 Qe8 30. Nxf7+ Rxf7 31. Rxe6 Qxb5 32. Rh6+ 1-0