In order to checkmate with a knight and bishop, one of the most common methods is the W Maneuver.
While not the only way to achieve checkmate, it’s the most instructive.
I’ll begin by starting with the ideal situation of the king on the edge of the board and in the wrong corner.
To view my in depth guide on tons of checkmate patterns, check out this article
1… Ke8 2. Ne5
What if the king doesn’t try to stay in the corner?
(2… Kd8 3. Ke6 Kc7 4. Nd7
The king seemingly gets out, but the bishop is in time to create a wall with the knight.
5… Kc7 6. Be4
Tightening the grip again and the king goes back and we continue the W maneuver.
6… Kc8 7. Kd6 Kd8 8. Bg6
Now the W maneuver continues
This makes things easier. White just plays a few moves to remake the W, but 2 files to the left.
3… Ke8 4. Ke6
4… Kd8 5. Kd6 Ke8 6. Bg6+ Kd8 7. Nc5 Kc8 8. Bf7 Kd8
9. Nb7+ Kc8 10. Kc6
10… Kb8 11. Kb6 Kc8 12. Be6+
The king is now stuck shuffling between a8 and b8. All you have to do is avoid stalemate
Now the white knight has to relocate
13… Ka8 14. Bf5
A tempo move with the bishop is needed
14… Kb8 15. Na6+ Ka8 16. Be4#
Now that we’ve learned what the W Maneuver is, let’s see how we get to it when the king is not conveniently stuck in the corner.
The key is to do it in as few moves as possible since you have 50 moves to complete the checkmate to avoid a draw.
In this set of moves, I completed this in 15 moves to make it as instructive as possible. I begin by getting the king to the edge of the board, and then to the “wrong corner” so I can push him to the right corner to deliver checkmate.
1. Ke2 Ke5 2. Kd3 Kf6 3. Ke4 Ke6
Begin by moving the king forward as much as possible, and then bringing the minor pieces into the action.
4. Bc4+ Kf6 5. Ne3 Ke7 6. Ke5
The king is now on the second rank
6… Kd7 7. Bd5
Blocking off all key squares and keeping the king on the second rank.
7… Ke7 8. Nc4 Kf8 9. Kf6
Now the king has moved to the first rank.
9… Ke8 10. Ne5 Kf8 11. Be4 Ke8
These next few moves will be for illustrative purposes to get the king to the wrong corner so that we can begin the W Maneuver.
12. Bf5 Kf8 13. Nf7 Kg8 14. Be4 Kf8 15. Bh7
We have now achieved the previous section’s diagram.
We could have pushed the king to the correct corner, but this will better illustrate the W maneuver.