knight and bishop checkmate.

https://www.chess.com/a/WzunQ91GUo34 take a look at this analysis

The important thing you must know is that the only area of the board where you can only force mate is in the corner with the same color as the bishop controls. If you have a light squared bishop for example, you must drive the enemy king to a8 or h1 to force mate. Now, the process of forcing the king to that area of the board and then delivering mate is broken down into the following steps:

Step 1: Force the defending king to the edge of the board

The idea is to take away as many central squares from the enemy king as possible and slowly force it back. Given that your rating is 1500+, this shouldn't be much of a challenge for you; you should be able to do it yourself even without any concrete advice on technique. Practice a few times and it should be easy, but I think that you would be able to wing this portion over the board in a non-blitz game even without practice.

Now, a good defender will run to the "wrong corner"; that is, the corner that your bishop doesn't control, because you can't force mate there. Here is an advice if you're the defender: try to stay near the center for as long as possible, and when finally forced towards the edge, run towards the corner that the bishops doesn't control. This maximizes your chance of holding out for 50 moves and claiming a draw if your opponent doesn't know what to do (you still do have to be somewhat careful even in the wrong corner since although they cannot be forced, checkmate positions do exist if you're careless - just as they do in 2 knights vs a king).

Once you keep the enemy king securely barricaded around the wrong corner (eg make sure that they can't escape back to the center), you start something known as the W-manuever. 

Step 2: Getting to the ideal position to start the W-manuever

For the rest of the post we will assume that the enemy king is on h8, you have the light squared bishop, and you want to get the enemy king to a8 (this is how I do it in drills). You can probably figure out the rest of the cases by rotating/flipping the board and using symmetry and memorizing the key ideas that I will highlight in bold.

Suppose you have kept the enemy king trapped around h8. Now, your goal is to get into this position to start the W-manuever, with black to move.

Note that the kings are in opposition and the knight is directly in between the kings and on the same color as your bishop. This is an important concept to remember in case your opponent doesn't defend like the computer and ends up closer to the a8 corner when driven to the edge, since the W-manuever still works if the kings are in opposition on d6 and d8 and your knight is on d7 since your bishop controls d7 (but it doesn't work if the kings are on e6 and e8 since the knight would be on e7, which is not controlled by your bishop).

The way you get into this position is by taking away squares from the enemy king. You want to get your king to f6 first when the enemy king is stuck around h8. Then, if your enemy king is on h8, then you first take the h7 square away from the enemy king, by moving your bishop to the b1 to h7 diagonal, or by making a waiting move if your bishop is already there (if the enemy king is on h7 then you take away the g8 square by moving your bishop to the a2 to g8 diagonal and the problem is the same by symmetry but flipped). This confines the enemy king to the top edge of the board, in which you are ready to move your knight to f7 and begin the W-manuever. Here is an illustration of this process, getting to the ideal position after you drive the enemy king to the wrong corner.

Step 3: The W-manuever

This step attempts to herd the defending king from h8 to a8. Our goal is to get into a position where the bishop controls c7 (from the b8 to h2 diagonal) and the king is on b6, confining the enemy king to a8 and b8. At this point we can prepare mate. The following diagram illustrates this position more clearly; note that you control the highlighted squares, which cut off the squares a8 and b8 from the rest of the board and thus trap the enemy king in the 2 squares a8 and b8 for the rest of the game.

The process is called a W-manuever because the idea is to move the knight in a W-shape, in the highlighted squares of the following diagram. Note that the final square b7 of the knight is the last highlighted square that the knight moves to. The way I remember this is that the knight belongs is same colored square as the bishop on the 7th rank (where the W-manuever typically begins) and opposite colored square as the bishop on the 5th rank.

Note that from the end of the earlier step the only legal move for black is Ke8. Now, you begin the W-manuever by moving your knight to e5, and at this point there are 2 possible defenses (you don't want to play something like Ke6 since the enemy could move back to f8, forcing your king back to f6 to guard h7, and you have made no progress).

Defense 1: The enemy king plays Kf8, trying to cling to the wrong corner

This defense actually loses quicker and is easier to play against, so I will illustrate how to win against this defense first in the following diagram.

If you think about which squares are taken away from the enemy king, you will get the motivation behind the W-manuever. Since you generally want your king in opposition (and thus in the 6th rank) to the enemy king to prevent them from running to the center of the board, the king cannot guard any squares on the 8th rank. The idea of the W-manuever is that the bishop cannot control dark squares on the 8th rank, and the king is in the 6th rank preventing the enemy king from escaping to the 7th rank, so we want to control dark squares on the 8th rank with your knight. This occurs when the knight is on the 7th rank on the same colored square as the bishop, which is what the W-manuever does. In summary, the W-manuever allows the knight to complement the bishop and take the maximum amount of squares on the 8th rank as possible away from the enemy king (while keeping your king in the 6th rank to keep the enemy king from escaping to the 7th rank).

Note a key concept: You generally want to keep the kings in opposition, but only if your opponent cannot escape towards the wrong corner, which is what happens when your enemy king is on a square on the edge controlled by your bishop. In that case, continue the W-manuever instead.

Defense 2: The defender plays Kd8, attempting to leave the edge of the board.

This defense lasts longer and is what the computer uses when you play against it. If you're the defender, you want to do this and run away from the edge when possible. This might catch your opponent off guard and even if they remember the W-manuever, they might not know what to do from here.

The idea here is to play Ke8 (getting closer to the enemy king) and continue the W-manuever even though you let the enemy king temporarily escape the edge of the board, but this is an illusion since the king is still confined in a small area in which it cannot escape, just not on the edge. The following diagram illustrates the steps here:

It looks like the enemy king has escaped after this since it was off the edge. I was confused at this point when trying the drills (before the key move Bd3!) because the previous video I watched didn't do a good job explaining what to do once the enemy king leaves the edge of the board. However, this is an illusion. Note that in the previous diagram, the enemy king is actually confined in a small region. The following diagram highlights the squares where the enemy king is confined after the last step.

Note that you control all of the highlighted squares, and they form a complete perimeter that cuts the a8 square off from the rest of the board (eg the enemy king can never escape the area since all paths to the area outside of the perimeter must cross one of the highlighted squares which would be put the enemy king in check and thus be illegal). This is how you should think about the key move Bd3! which seems random at first (you should think about this move as forming a complete perimeter in which the king can't escape; this is especially true when the board is rotated or flipped as you it is hard to memorize the equivalent d3 square in all cases). This bishop move is very important and you should spend time to really digest the idea of creating a confinement "net" for the enemy king.

The next step is to take away the c6 square from the enemy king and tighten the perimeter, and eventually get to the key position to prepare checkmate. Note that in most cases the enemy king will eventually move to d8, and you will get into the same position as the one where you started the W-manuever but 2 files closer to the right corner, and can repeat the same concepts except that you don't have to worry about the king escaping the edge of the board (since the king would be on the a file rather than the c file by the time it escapes the 8th rank and very close to the corner, no further down than a6 so completing the W manuever with Nb7 would keep the enemy king confined on a6 to a8 and this would essentially be the board rotated).

Below is an illustration if the enemy attempts the same idea as before, trying to escape the 8th rank with Kb8 (if you're trying to defend you should generally play Kd8 here since it holds out longer; you can't really escape the edge here so your opponent will likely know what to do anyway. The difference here is slim however unlike the Kd8 vs Kf8 choice earlier, and it's unlikely that this will make the difference between the 50 move cutoff if your opponent has made it this far).

Step 4: Checkmate

Now you've made it so close, you want to bring home the full point! At this juncture your king and bishop together confine the enemy king to the squares a8 and b8 and force them to repeat the moves Ka8 and Kb8. The idea is to eventually bring your knight to a6 with check, taking away the b8 square from the enemy king, and then delivering checkmate by moving your bishop to the a8 to h1 diagonal for the hard earned point. Note that you do not want to take away the b8 square by moving your knight to d7 since that blocks your bishop on the c8 to h3 diagonal, and allows the enemy king to thus run to c8.

Note that you do not want to bring your knight to a6 without check, since that would be stalemate! You did not come this close only to throw away half a point! In a case where your knight could move to a6 next move but the enemy king is on a8 (and thus Na6 would not be check), you make a waiting move by moving your bishop along the c8 to h3 diagonal, maintaining control of the crucial c7 square and preventing the enemy king from escaping a8-b8. Here is an illustration of the final touch on the board.

For me, this process usually works in 30-40 moves (quick enough to avoid a draw by the 50 move rule), and even in the toughest position (where the win takes 33 moves with best play from both sides), it took me 41 moves to checkmate the computer, which is still relatively comfortable given that most positions are easier to win than the toughest position and that most humans defend worse than computers (and that the number of moves between variations doesn't differ that much.

Congratulations if you have made it this far! Since this endgame occurs 1 out of 6000 times, and you will be the superior side about half the time, this means that you will be able to win 1 out of 12000 times more than before (and maybe the gain is little bit more because if you're the defender, you now know to head to the wrong corner if forced to the edge successfully and hope your opponent does not know this technique. Also, you might still have been able to win this sometimes even before reading this due to luck or bad defense), so assume that about 1/10000 you now turn a draw to a win or a loss to a draw from reading this. This means that you gain about 1/20000 expected outcome, which means corresponds to a rating gain of 1/30 (I honestly expected this endgame to occur more frequently, due to the enemy sacrificing their last pieces for your last pawns).