How do you checkmate with the pair of bishops? How do you checkmate with the pair of Bishops and King versus King?

Are there any stalemates to look out for?

What are the key ideas / positions / tricks to make it work?

Thanks!
I always like to explain this in a visual way. 

Basic Idea: Keep the bishops together. They form a large net (restricted area) from which the opponent king cannot escape. 



Step 1: Push Opponent's King To Edge Rank or File

Keeping the bishops together and using the king for support, make the restricted area smaller to push the opponent's king back to an edge rank or file. 

     [FEN "8/8/8/3k4/8/8/8/K2BB3 w - - 0 1"]

    1. Bd2 Ke4 2. Be2 Kd5 3. Kb2 Ke4 4. Kc3 Kd5 5. Bd3 Ke5 6. Be3 Kd5 
    7. Bd4 Kd6 8.Be4 Ke6 9. Kc4 Kd6 10. Bd5 Ke7 11. Be5 Kd7 12. Kc5 Ke7 
    13. Kc6 Kf8 14. Bf6 Ke8 15. Be6 Kf8  


1. Bd2 Ke4 2. Be2

 

2...Kd5 3. Kb2 Ke4 4. Kc3 Kd5 5. Bd3 Ke5 6. Be3



6...Kd5 7. Bd4 Kd6 8.Be4 



8...Ke6 9. Kc4 Kd6 10. Bd5 Ke7 11. Be5



11...Kd7 12. Kc5 Ke7 13. Kc6 Kf8 14. Bf6 Ke8 15. Be6 Kf8 (note: I intentionally chose the 13...Kf8 line because that is more instructive. For the 13...Kd8 line, please see the sample mates provided at the end).



Step 1 Accomplished! 

Step 2: Push the Opponent's King to a Corner

In the above position, we shall choose the h8 corner. The key is to keep the opponent's king on the back rank while slowly taking away squares from the other side. 

The idea now is to move the king over to the right hand side via Kc6-d5-e5-f5-g6 to cover f7 and then move the bishop to d7 to cover e8, thereby pushing the opponent's king to the corner. 



   [FEN "5k2/8/2K1BB2/8/8/8/8/8 w - - 0 16"]

   1. Kd5 Ke8 2. Ke5 Kf8 3. Kf5 Ke8 4. Kg6 Kf8 5. Bd7 Kg8


Now, we can take control over f8, thereby pushing the opponent's king to the corner. 



   [FEN "6k1/3B4/5BK1/8/8/8/8/8 w - - 0 21"]

   1. Be7 Kh8


Step 2 Accomplished! 

Step 3: Checkmate the Cornered King

Here, it is important to be careful to avoid a common stalemate that occurs in this position. 



We observe that White has to make a waiting move in order to challenge the g8 square with check, so that then after Kh8, Bf6+ would be mate. 



And now the mate is easy. 

    [FEN "7k/3BB3/6K1/8/8/8/8/8 w - - 0 22"]

    1. Be8 Kg8 2. Bf7+ Kh8 3. Bf6#


Below, I have included other types of checkmates that can occur if Black play's differently. I've also included the fastest mate from the original position which starts with 1.Bc3! 

    [FEN "8/8/8/3k4/8/8/8/K2BB3 w - - 0 1"]

    1. Bd2 (1. Bc3 Ke4 2. Kb2 Kd3 3. Bf3 Ke3 4. Bd5 Kd3 5. Be5 Kd2 (5... Ke3) 6.
    Bd4 Kd3 7. Bc5 Ke2 8. Kc2 Kf1 9. Kd3 Ke1 10. Bg2 Kd1 11. Bf2 Kc1 12. Kc3 Kd1
    13. Bf3+ Kc1 14. Be3+ Kb1 15. Kb3 Ka1 16. Bd1 Kb1 17. Bc2+ Ka1 18. Bd4#) 1...
    Ke4 2. Be2 Kd5 3. Kb2 Ke4 4. Kc3 Kd5 5. Bd3 Ke5 6. Be3 Kd5 7. Bd4 (7. Bf4 Kc5
    8. Be4 Kb5 9. Be3 Ka6 10. Kc4 Ka5 11. Bc6 (11. Bb7 Ka4 12. Bb6 Ka3 13. Kc3 Ka4
    14. Bc6+ Ka3 15. Bc5+ Ka2 16. Kc2 Ka1 17. Ba4 Ka2 18. Bb3+ Ka1 19. Bd4#) 11...
    Ka6 12. Bc5 Ka5 13. Kd5 Ka6 14. Kd6 Ka5 15. Kc7 Ka6 16. Bb4 Ka7 17. Bb5 Ka8 18.
    Ba5 Ka7 19. Bb6+ Ka8 20. Bc6#) 7... Kd6 8. Be4 Ke6 9. Kc4 Kd6 10. Bd5 Ke7 11.
    Be5 Kd7 12. Kc5 Ke7 13. Kc6 Kf8 (13... Kd8 14. Bd6 Ke8 15. Be6 Kd8 16. Bf7 Kc8
    17. Be7 Kb8 18. Kb6 Kc8 19. Be6+ Kb8 20. Bd6+ Ka8 21. Bd5#) (13... Ke8 14. Bd6
    Kd8 15. Bf7 Kc8 16. Be7 Kb8 17. Kb6 Kc8 18. Be6+ Kb8 19. Bd6+ Ka8 20. Bd5#) 14.
    Bf6 Ke8 15. Be6 Kf8 16. Kd5 Ke8 17. Ke5 Kf8 18. Kf5 Ke8 19. Kg6 Kf8 20. Bd7 Kg8
    21. Be7 Kh8 22. Be8 Kg8 23. Bf7+ Kh8 24. Bf6# *