A tough mate, probably one of the toughest. The opponents king must be pushed towards the corner which is the same color as the color your bishop sits on. If you are the one with only a king left always head towards the corner opposite to the bishops color. Most will give up and allow a draw.

I love this mate because it is so fragile.

 --- --- --- ---
|   |   |   |BK |
 --- --- --- ---
|   |   |   |   |
 --- --- --- ---
|   |   |WK |WN |
 --- --- --- ---
|WB |   |   |   |
 --- --- --- ---  

Take your time with this ending. First bring your opponent's King to one of the board's edges. Second drive the King to the appropriate corner. Keep the knight and king beside each other (this prevents your opponent from attacking the knight) and cut off your opponents path away from your object corner. Use a well timed zugzwang to force your opponent to make moves he/she would rather not make. Use the bishop to deliver the checks and ultimately the checkmate.

I have never seen this ending occur. I came across it while going through endgames and liked it. I use it as a good drill for understanding coordination between a bishop and knight or king and knight.

Endgames in Chess
Chess Openings
Chess

Bishop, Knight, and King vs. King - A Chess Endgame

It is very unlikely that you will ever be in a situation thar requires mate with only a bishop and knight, but it's good to know nonetheless. Personally, I've never been on the offensive side of this endgame, but I was once on the defensive. Luckily, my opponent didn't know how to execute it correctly; and after a few moves of which I was able to elude, he offered a draw. Hey, it's better than a loss.

Before I get into the meat and potatoes of this endgame, I'd like to cover a few key points. First, you're going to mate your opponent in a corner that is the same color of the square that your bishops resides on. If you have a bishop on a light square, then you are going to mate the king on h1 or a8 (both light squares). Conversely, if your bishop resides on a dark square, you're going to mate the king on a1 or h8 (both dark squares). Second, you will need the involvement and careful planning of all three of your pieces to drive the king into the corner for the mate. Third, your opponent may make things difficult for you by retreating to a corner that is opposite to the one you need him in. Don't fret this, because once you trap him in that corner, you can drive him to other.

So, on to the example. Here is our imaginary setup:

BTW, take a look at chess notation if you are a complete newbie to chess. But, just in case you are lazy...

BK = Black King
WK = White King
WB = White Bishop
WN = White Knight

+---+---+---+---+---+---+---+---+
|   |   |   |   |   |   |   |   | 8
+---+---+---+---+---+---+---+---+
|   |   |   |   |   |   |   |   | 7 
+---+---+---+---+---+---+---+---+
|   |   |   |   |BK |   |   |   | 6
+---+---+---+---+---+---+---+---+
|   |   |   |   |   |   |   |   | 5
+---+---+---+---+---+---+---+---+
|   |   |WN |WK |WB |   |   |   | 4
+---+---+---+---+---+---+---+---+
|   |   |   |   |   |   |   |   | 3
+---+---+---+---+---+---+---+---+
|   |   |   |   |   |   |   |   | 2
+---+---+---+---+---+---+---+---+
|   |   |   |   |   |   |   |   | 1
+---+---+---+---+---+---+---+---+
  A   B   C   D   E   F   G   H

Okay, in our example, white's pieces are grouped together in the middle. Since the bishop is on a light square (e4 is a light square), we have to drive the king to either a8 or h1 for the mate. The king is closest a8, so we'll go for that one. There is really no exact formula for this process. Getting the king trapped in the corner can be tough. The only advice I can give you is to try to keep your minor pieces (knight and bishop) in the middle while you shepherd with your king. Once the opponent is in the general vicinity of the corner, then it will be time to use the minor pieces.

1 Kc5 Kf6 - And we're off.
2 Kd5 Kf7 - Black's king looks like he wants to occupy the corner we don't want him in.
3 Ke5 Kg7 - Yup, black is going to h8.
4 Ke6 Kf8 - Almost there.
5 Kf6 Kg8 - Now it's time to put those minor pieces to use.
6 Ne5 Kh8 - Good, he's in the corner. Time to drive him the other way.

From the beginning, black stayed around the area of h8. White's king was able to shepherd him into just a few squares of the corner. When we started to bring the minor pieces up, instead of going to the corner we want him in, black retreats to the other side.

+---+---+---+---+---+---+---+---+
|   |   |   |   |   |   |   |BK | 8
+---+---+---+---+---+---+---+---+
|   |   |   |   |   |   |   |   | 7 
+---+---+---+---+---+---+---+---+
|   |   |   |   |   |WK |   |   | 6
+---+---+---+---+---+---+---+---+
|   |   |   |   |WN |   |   |   | 5
+---+---+---+---+---+---+---+---+
|   |   |   |   |WB |   |   |   | 4
+---+---+---+---+---+---+---+---+
|   |   |   |   |   |   |   |   | 3
+---+---+---+---+---+---+---+---+
|   |   |   |   |   |   |   |   | 2
+---+---+---+---+---+---+---+---+
|   |   |   |   |   |   |   |   | 1
+---+---+---+---+---+---+---+---+
  A   B   C   D   E   F   G   H

Now it's time to drive your opponent to the other corner. Again, there is no formula for this. Though, one thing you must do is make moves that don't actually threaten anything. That is, moves that put the king in check will often be more harmful, as he will most likely not be traveling the way you want him to. So, you must place your pieces so that the king has no choice but to move into the direction he does not want to go. This is called zugzwang.

7 Nf7 Kg8 - Force the king left.
Note: Black could have either gone to g8 or h7, but it doesn't matter. Going to h7 means he would be forced into h1, the other desirable corner.
8 Bf5 Kf8 - As I discussed, this was a good, non-threatning move.
Note: If white would have gone for the check at Bd5, that would have done nothing to improve the situation.
9 Bh7 Ke8 - White's bishop cuts off the retreat of black and forces him left.
10 Ne5 Kd8 - White's knight prevents the king from leaving the back rank and forces him further left.
11 Ke6 Kc7 - Bring the king over so he can support the next move of the knight.
12 Nd7 Kc6 - Prevent the king from moving to b6.
13 Bd3 Kc7 - Another good, but anti-climatic, move which prevents escape.

And here we are. The black king is well on his way to the favorable corner. Remember, this part of the mate is very fragile. Not only should you look for the move that will drive the king further into the corner, but also look ahead to make sure you will be able to further shepherd him.

+---+---+---+---+---+---+---+---+
|   |   |   |   |   |   |   |   | 8
+---+---+---+---+---+---+---+---+
|   |   |BK |WN |   |   |   |   | 7 
+---+---+---+---+---+---+---+---+
|   |   |   |   |WK |   |   |   | 6
+---+---+---+---+---+---+---+---+
|   |   |   |   |   |   |   |   | 5
+---+---+---+---+---+---+---+---+
|   |   |   |   |   |   |   |   | 4
+---+---+---+---+---+---+---+---+
|   |   |   |WB |   |   |   |   | 3
+---+---+---+---+---+---+---+---+
|   |   |   |   |   |   |   |   | 2
+---+---+---+---+---+---+---+---+
|   |   |   |   |   |   |   |   | 1
+---+---+---+---+---+---+---+---+
  A   B   C   D   E   F   G   H

14 Bb5 Kd8 - Prevent the move to c6 and force the king further left.
15 Nb6 Kc7 - Move out of the way so the bishop can cover the diagonal that prevents escape.
16 Nd5 kb7 - And we're very close to mate. Black is now trapped in a box in the light colored corner. It is now our job to make that box smaller.

+---+---+---+---+---+---+---+---+
|   |   |   |   |   |   |   |   | 8
+---+---+---+---+---+---+---+---+
|   |BK |   |   |   |   |   |   | 7 
+---+---+---+---+---+---+---+---+
|   |   |   |   |WK |   |   |   | 6
+---+---+---+---+---+---+---+---+
|   |WB |   |WN |   |   |   |   | 5
+---+---+---+---+---+---+---+---+
|   |   |   |   |   |   |   |   | 4
+---+---+---+---+---+---+---+---+
|   |   |   |   |   |   |   |   | 3
+---+---+---+---+---+---+---+---+
|   |   |   |   |   |   |   |   | 2
+---+---+---+---+---+---+---+---+
|   |   |   |   |   |   |   |   | 1
+---+---+---+---+---+---+---+---+
  A   B   C   D   E   F   G   H

17 Kd7 Kb8 - This confines the box to four squares.
18 Ba6 Ka7 - Now it's only three squares.
19 Bc8 Kb8 - Move the bishop out of danger, but still retain the box.
20 Ne7 Ka7 - Set up the knight so that he may move to c6 and eventually deliver the mate.
21 Kc7 Ka8 - The box consists of only two squares now.
22 Bb7+ Ka7 - One more move!
23 Nc6++ - And that's mate.

+---+---+---+---+---+---+---+---+
|   |   |   |   |   |   |   |   | 8
+---+---+---+---+---+---+---+---+
|BK |WB |WK |   |   |   |   |   | 7 
+---+---+---+---+---+---+---+---+
|   |   |WN |   |   |   |   |   | 6
+---+---+---+---+---+---+---+---+
|   |   |   |   |   |   |   |   | 5
+---+---+---+---+---+---+---+---+
|   |   |   |   |   |   |   |   | 4
+---+---+---+---+---+---+---+---+
|   |   |   |   |   |   |   |   | 3
+---+---+---+---+---+---+---+---+
|   |   |   |   |   |   |   |   | 2
+---+---+---+---+---+---+---+---+
|   |   |   |   |   |   |   |   | 1
+---+---+---+---+---+---+---+---+
  A   B   C   D   E   F   G   H

So, Remember:

• You will mate the king in a corner that has the same color square as the bishop.
• Zugzwang is very important in this endgame.
• Don't worry if your opponent flees to the other side. Trap him in that corner and drive him to the other!
• When trapped in the proper corner, make the box smaller!
• This is a hard mate, probably the hardest. Don't get discouraged if you mess up. Most people will simply draw at this point anyway.