The ending of rook and pawn vs bishop is usually an easy win for the side with the rook.
The side with the rook has to activate the rook as well as the king.
I'll show you how it's done.
In this position White is winning. The standard winning method goes like this.
Now Black can't play 1...Kd6; because this would give you the possibility to liquidate into a won pawn ending with 2.Rxc7, Kxc7; 3. Ke5. (If this doesn't sound familiar to you, you may want to check out the basic king and pawn endgame).
Thus Black plays something like
Now you'll be able to activate your king and rook.
Now Black has to decide if he wants to retreat his king or interpose the bishop.
This gives us two variations to examine. 2....Kd7 and 2....Bd6.
Blakc retreats his King with 2...Kd7;
This gives your king the possibility to enter the scene of the action with:
The further your king advances, the easier you'll get the pawn to the other side.
Now the bishop is attacked and has to move. This gives your king the oppotunity to advance even further.
4...Bf4; 5. Kc6,
Now that your king has come this far, you'll be able to advance the pawn without any problems.
Black tries to keep his king in place with
Unfortunately this doesn't help, because after
The black king has to leave its optimal defending post, and your king enters the scene.
3.... Ke7; 4.Kd5,
The black bishop is forced to move and your king keeps moving forward.
5....Be3; 6.d5, Bf4; 7.Ra7+,
The black king has to move away again. Once the black king no longer controls the d6-square, your pawn will move forward unhindered.
The endgame of rook and pawn vs bishop should be an easy win for you.
You'll activate your rook as well as your king. This clears the path for your pawn.