Bishop and King versus Rook and King is a draw. Yet it is easy to go wrong and lose. How do you hold a draw?
Examples and instructions are taken from the book:
Y.Averbakh - Comprehensive Chess Endings Volume 2.
RELEVANT POSITIONS FOR PLAYING THIS ENDGAME SUCCESSFULLY:
I will quote the introduction from the above book:
"It is impossible to win against correct defense, even if the defender's king is driven back to the edge of the board. When retreating under the pressure of the opposing pieces, all he must do is to reach the corner of the opposite color to the bishop. We will call this the safe corner".
[Title "Weaker side holds balance"] [fen "7k/R7/7K/8/8/1b6/8/8 w - - 0 1"] 1.Ra8+ Bg8=
White can't improve his position, since moving his king or moving rook along the 8th row will lead to stalemate. Black maintains balance by keeping his bishop on
Again, quoting the same resource:
"If the defender's king is in a corner of the same color as his bishop, a win is normally possible. We will correspondingly call this the dangerous corner."
The following position demonstrates winning method when weaker side has king in "the dangerous corner":
[Title "Horwitz & Kling, 1851"] [StartFlipped "0"] [fen "6k1/5R2/6K1/8/3b4/8/8/8 b - - 0 1"] 1...Bg1! ( 1...Bc5 2.Rc7 Bf8 3.Rc8+-) 2.Rf1 Bh2! 3.Rf2 Bg3 4.Rg2! Be5 ( 4...Bh4 5.Kh5++- ) ( 4...Bd6 5.Rd2 Be7 6.Rc2!+- ) 5.Re2 Bd6 6.Re8+ Bf8 7.Rd8+-
Black must hide his bishop as in the main line, and we saw how to flush it out and win. The sub-variation shows what happens when he refuses to do so.
Here is another important position, where the result depends on who's turn to move is:
[Title "Horwitz & Kling, 1851"] [fen "5k2/6R1/5K2/8/8/5b2/8/8 w - - 0 1"]
With the kings in opposition mating threats are in the air and if White moves first he is able to win. For the moment bishop is hiding behind the opposing king, but with the above rook maneuver White can flush it out:
[Title "Horwitz & Kling, 1851"] [fen "5k2/6R1/5K2/8/8/5b2/8/8 w - - 0 1"] 1.Rg3 Be4 ( 1...Bh5 2.Rh3 Bf7 3.Rh8+ Bg8 4.Kg6+- ) ( 1...Bc6 2.Rc3 Bd7 3.Rb3! Kg8 4.Rb8+ Kh7 5.Rb7+- ) 2.Re3 Bg2 3.Re2! Bf3 4.Rf2+-
But if it is Blacks turn to move he saves the game by running with his king:
[Title "Horwitz & Kling, 1851"] [StartFlipped "0"] [fen "5k2/6R1/5K2/8/8/5b2/8/8 b - - 0 1"] 1...Ke8! 2.Ke6 Kd8 3.Kd6 Kc8=
c6 square is inaccessible to the White king he is unable to make any mate threats.
These are the basic positions required to know in order to play this endgame well.
In the above book were provided studies which were complex, but can be solved with the knowledge of the above covered positions.
To keep this post clear, concise and useful I will omit them.
First answer, I am hoping that there will be better answers with nice diagrams
Your goal should (could) be to get your king to a corner that is not the colour of your bishop. Your friend stalemate welcomes you there. White can chase you out of the corner, but not without letting you escape to center and eventually into the corner opposite the original corner. Of course if you enter the ending in awkward manner, for example with your king on the edge of the board, some other tactics/strategy may save your life.
This site explains the defence in the corner quite well.
Elsewhere on the same site there is also a tactic for the party with the rook to win if you are near the wrong corner. If you are on the edge of the board, but not near a corner, I think there are cases where you have to play very carefully/accurately.