I can't do this mate! I was just now playing against much lower rated opponent than me, and he ended up trading his last piece for my last pawn. In silman's endgame course silman says that it's so rare that there's no point in studying it but it's now happened twice so I think it would be worth the time to know how to win these. Can someone explain the technique?

The basic idea is to get the king out of the center with your pieces (like all of the  basic mates in endgames), and the opponent's king is going to try and go to the opposite color of the bishop you have, and you try to get them on the same color bishop you have to be able to checkmate.

http://en.wikipedia.org/wiki/Bishop_and_knight_checkmate

http://www.chessvideos.tv/forum/viewtopic.php?f=12&t=802

if it's a rapid game your in a bit of a fix, if it's an online game i'm sure your logic can figure it out.

Don't feel too bad...it's pretty tough. It took me a long time before I felt like I could do it with any degree of confidence (and even then, it takes time--I mean, on the clock). I once saw an IM in an online speed game hang one of the pieces deliberately, thus acknowledging that he couldn't do it fast either (so it's even beyond someone of that calibre). I suppose Karpov could do it in a 1-minute game!--but then, that's why he's one of the gods of chess.

This ending is quite rare, but very difficult to master. I am still getting used to it myself, although I am now pretty sure I can mate with it. The only way to learn it is to keep practicing. Remember, you must mate in 50 moves! (A great book on this and other endings is Winning Chess Endings by Yasser Seirawan.)

...and that's the really hard part. Several times I've trained myself to do this mate somewhat reliably but without practice the skill goes away. One slip and the enemy king can slip between your pieces and gallop across the board, eating precious moves. It usually took me around 35 moves, when I could do it at all.

I learned using "Basic Chess Endings" by Reuben Fine and lots of practice against a computer. I'm sure there must be something better out there by now.

That is a strange viewpoint I would say, because you can learn a lot about playing with a knight and a bishop in combination, thus you can benefit from such a study whether you encounter an endgame of that type or not in your games. It is not for beginners though, I think I would regard it as part of basic chess study once you have been through the beginners stuff.

I used to be of the opinion that, since the mate was both a) hard and b) rare, there was no point. But not that long ago I decided that this was a ridiculous viewpoint--it is, after all, a basic mate, and it's embarrassing not to know how to do it. I may never need to put it together in a serious game, but that's hardly an excuse not to study it.

So I watched some videos and set up the position against my computer, then drilled it over and over again. It wasn't long before I found myself very comfortable with it--I now know I can do it, and in fact I can do it quickly. There are really only a couple of positions that are difficult; but once you work them out you'll never stumble again.

Does this knowledge make me a better chessplayer? Probably not. But if someone ever asks me if I can do the B+N mate, I won't have to hang my head in shame.

I just did some poking around and found this nice demonstration.

http://www.chess.com/article/view/how-to-checkmate-with-king-bi

Here is a quick example showing part of the process from the point when the enemy king is in the wrong corner. Pay special attention to the position after white's 5 Kf5, since it's the trickiest in this mate, and I suspect it's the point at which most people believe they've failed.

As I said in my former reply, you can learn a lot about playing with a knight and a bishop in combination, and thus benefit from such an endgame study whether you encounter an actual endgame of that type or not in your games.

How so, BlueKnightShade? It seems hard to benefit from this other then being able to win it if it happens because this is totally different, than, for example, opening moves or middlegame planning. The answer "to see how they work together" doesn't help actual skill and it is probably too difficult to be worth it.

However, DimKnight's excellent tutorial makes it seem easy to me, but with no help that's probably not the case!

I'm somewhere in the middle of the DlueKnightShade/Elubas position. Working through the B+N mate probably does help improve your basic chess skillset, in that one needs to appreciate how these two minor pieces can collaborate in building walls. One might also argue that working through the two-bishop mate helps you to appreciate the power of the "two bishops." But perhaps a chessplayer would be better served by studying more common endgames.

In the end, though, the B+N mate is not that hard; and if you have an hour or so in an afternoon you can probably learn it well enough. So why not give it a whirl?

[And, while I thank you Elubas, I didn't set out to create a tutorial!]

I agree with BlueKnightShade, any practice with piece coordination is good practice. You may not be trying for a mate but you could easily be trying to queen a pawn with a bishop and knight against a rook (or more). Being familiar with how a bishop and knight work together could make all the difference.

DimKnight, I also liked your example. That one tricky move is probably where I've messed up lots of times.

It's a pain to do.  Simple but never easy.  Before the B+N, you need to be comfortable using the K as an offensive piece.  It is a very strong piece in many end games. 

But, for this, first you need to use the King and the pieces to force the opponent's king to the edge. Then escort it to the corner that is the same color squares as the bishop.  And, make sure while containing the King that you don't set up a stalemate situation. 

As I said above, simple but simple doesn't equate with easy.  In a blitz game, I might just offer the draw.

 In blitz even if you run out of time the game will be drawn if all your opponent has is a king so why offer a draw ?  If he flags he loses.