When is a bishop stronger than a rook? Usually the rook is stronger than the bishop. Are there situations (and obviously I'm not assuming equal material) in which the latter is stronger?
The standard value for a piece that we learn early on (Queen=9 pawns, Rook=5 pawns, Bishop,Knight=3 pawns) is often a very good guidepost for people to make proper trades and find good tactical sequences. However, it should not be taken for granted that a rook will always be better than a bishop. Let's look at one extreme example:

       [fen "rn5k/3p4/p1pP4/PpP5/1P6/8/5B1P/7K w - - 0 1"]

       1.h4 {White wins by using the h-pawn as bait for the black king, and then infiltrate on black's queenside with the white king.} Kh7 2.Kg2 Kg6 3.Kf3 Kf5 4.Be3 Ra7 5.h5 Ra8 6.h6 Kg6 7.Ke4 Ra7 8.Ke5! {White places the king as close to the queenside as possible before sacrificing the h-pawn.} 8...Ra8 9.h7! {Luring the black king away from the defense of the f6 square.} 9...Kxh7 10.Kf6 Kg8 11.Ke7 Kg7 12.Kd8 Kf8 13.Kc8 Ke8 14.Kb7 Kd8 15.Kxa8 Kc8 16.Ka7 {Zugzwang! Black's king has to move, which means that the knight is lost.} Kd8 17.Kxb8 Ke8 18.Kb7 Kd8 19.Kxa6 Kc8 20.Ka7 Kd8 21.Kb8 {and white will win by promoting the a-pawn.}


Here white's bishop was stronger than black's rook and knight together, and it's not difficult to realize why; the knight was completely paralyzed, and the rook couldn't get out because of that. Black's pieces were nothing more than a couple of sitting ducks, so to speak.

In general, a piece is only good if it is performing some specific task on the chessboard, such as attacking weaknesses in the enemy camp or defending weak points in one's own camp. If the piece just sits on the edge of the board like a bystander (as in the example above) then it's totally worthless.

The rook is often a very good piece; since it's a long range piece, it can often attack and defend at the same time, and it can easily switch targets if there are many open ranks and files for the rook to move along. The same can be said about the bishop, using diagonals instead of rank and files, but the bishop has a clear disadvantage when compared with the rook: it can only influence squares of one color! Let's look at an example where the rook dominates the bishop, from Grischuk-Radjabov, Corus 2003:

    [fen "4k3/5R2/4K3/8/5p2/5P2/3b4/8 w - - 0 1"]

    1.Rc7! {White threatens checkmate in one move, which forces black's king to move.} 1...Kf8 (1...Kd8?? 2.Rd7+ {picks up the bishop for white.}) 2.Kf6 {Threatening mate in one move again. Note how white's rook covers c3 so that the bishop cannot check the white king.} 2...Ke8 (2...Kg8 3.Rg7+ Kf8 (3...Kh8? 4.Kg6 {and black is getting mated shortly.} 4...Bb4 5.Rc7 Kg8 6.Rc8+ Bf8 7.Ra8! {Zugzwang! Black's king has to move.} 7...Kh8 8.Rxf8#) 4.Rd7! {The white rook is hitting the black bishop, as well as it threatens mate in one yet again.} 4...Bc3+ 5.Kf5 {and white wins, since the black bishop is unable to defend the pawn on f4 due to the white rook covering the d2 square.}) 3.Re7+ Kd8 4.Re4! {White's rook cuts off the black king from the kingside, and therefore Radjabov decided it was time to resign in this position.} (4.Re4 Kd7 {For the sake of completeness, this is how white wins if black plays on:} 5.Kf5 Kd6 6.Rxf4! Bxf4 7.Kxf4 Ke6 8.Kg5! {and white has achieved a well known winning pawn ending.})1-0


In this example we saw that the rook can make multiple threats at the same time and take away squares from the bishop. This because the rook had many open ranks and files to work on, which made switching targets very easy to accomplish for the rook. If the rook is tied down and restricted, it becomes way less impressive, as can be clearly seen in the following example:

   [fen "1r6/2p1p1k1/p1B1P1p1/Pp3pPp/1P3P1P/2P5/5K2/8 w - - 0 1"]

   1.Ke3 Rd8 2.Bd7 {White doesn't let black get any activity whatsoever.} 2...Kf8 3.Kd4 Rb8 {There isn't much black can do.} (3...Rxd7+ 4.exd7 e5+ 5.fxe5 Ke7 6.e6 {is just a lost pawn ending.}) 4.Kc5 Ra8 5.Kd5! {Of course there are other ways to win, such as breaking with the c3 pawn to create a passer, but this nice use of zugzwang forces black to give up the c7 pawn without a fight.} 5...Ra7 6.Kc6 {and black's king has to move away from the e8 square due to zugzwang.} 6...Kg7 7.Bc8 Kf8 8.Bb7 Ke8 {Black's king can't get to d8 in time to defend c7.} 9.Kxc7 {and black will lose everything.}


Here black has an open file for the rook, but the rook is kind of dominated by the extremely influential bishop on c6, who can block the rook's path whenever black plays ...Rd8 by placing itself on d7, while at the same time keeping the black king imprisoned on the kingside. White wins by walking the king up the board to attack the weak pawn on c7, and there is little black can do about it besides watch in horror as the position starts to crumble.

Edit: I finally managed to think of an improved example of a bishop being stronger than a rook.