What is counterplay? I hear this term in online commentaries. 

I have a vague sense of what it means: you earn yourself some good moves by ignoring a portion of the board and focusing on more promising parts of your play. It's as if you are telling your opponent: Sure you are threatening, but I am going to ignore it for I can also threaten you. So the players now try to prove whose threat is stronger.

Consider Anand-Andreikin for example. Andreikin found a draw by evaluating that his queen side was stronger. He chose to ignore Anand's strong king side. Or Anand-Carlsen, where Anand found a draw by finding counterplay with 44. Qh1. I am looking for heuristics/ rules of thumbs in this theme of playing.

I feel my understanding of the term is still loose. So:


Could you explain what is counterplay with some instructive example?
How to evaluate the potential of counterplay, i.e. - how to decide under practical playing conditions whether to counter attack or respond to the threat?
I feel the term is related to the idea of 'complicating a position'. Is it? (I have Nakamura-Carlsen on my mind, but it may not be the best example).
I have also heard of players talking of prophylaxis to prevent potential counterplay. Again can you give some heuristics to decide, whether to get prophylactic in a winning position or not? This is the same question as 2, except from the other player's point of view.


Thanks!
This is actually quite a sensible question and it isn't about the English word 'counter' as JiK suggests, but about the English word 'play'. ;-)

To have 'play' in a position basically means that you are able to create threats. The possibility to create threats isn't really linked to how complicated a position is. But a position in which only one player can create threats usually isn't very complicated. 
So there are positions with counter play that aren't very complicated, but going for counter play often complicates a position. 

'counter' just means that your opponent had all the play before and only now you start to strike back. 
You could say that 'counter play' is a defensive technique. You may not be able to parry your opponents threats, but by creating counter threats you can dissuade him from executing his threats. 

I'll try to come up with some endgame example later …

No endgame, but very recent and a really nice fight - Adams-Caruana:

rnbqkbnr/pppppppp/8/8/8/8/PPPPPPPP/RNBQKBNR w KQkq - 0 1

1. e4 e5 2. Nf3 Nc6 3. Bb5 a6 4. Ba4 Nf6 5. O-O Be7 6. Re1 b5
7. Bb3 O-O 8. d3 d6 9. c3 Na5 10. Bc2 c5 11. Nbd2 Nc6 12. Nf1
h6 13. Ne3 Re8 14. a4 Be6 15. h3 Bf8 16. Nh2 b4 17. Bb3 Rb8
18. Bc4 bxc3 19. bxc3 d5 20. Bxa6 Qa5 21. c4 Qxa6 22. exd5
Bxd5 23. cxd5 Nb4 24. Ra3 Nfxd5 25. Neg4 f6 26. Nf1 Rbd8
27. Ng3 c4 28. Nxh6+ gxh6 29. Qg4+ Kh8 30. dxc4 Ne7 31. Qh5
Ng8 32. Nf5 Qb7 33. Rg3 Qh7 34. Nh4 Ne7 35. Ba3 Nbc6 36. Kh2
Bg7 37. Bxe7 Nxe7 38. Rd1 Qg8 39. Rb1 Rb8 40. Rd1 Red8 41. c5
Rxd1 42. Qxd1 Rb4 43. Qd7 Rxh4 44. Qxe7 Rd4 45. c6 Qf8 46. Qb7
Rb4 47. Qd7 Rd4 48. Qb7 Rb4 49. Rc3 f5 50. Qd7 Rd4 51. Qe6 Rd6
52. Qxd6 Qxd6 53. c7 e4+ 54. g3 Bxc3 55. c8=Q+ Kh7 56. Qxc3 f4
57. gxf4 Qxf4+ 58. Qg3 Qd2 59. Qc7+ Kg6 60. Qb6+ Kh7 61. Qb7+
Kh8 62. Qa8+ Kg7 63. Qxe4 Kf6 64. Qf3+ Kg6 65. Kg2 Qa2
66. Qe4+ Kf6 67. Qf4+ Kg6 68. Qd6+ Kg7 69. Qe5+ Kh7 70. a5
Qg8+ 71. Kh2 Qf7 72. Qe4+ Kg7 73. a6 1-0


If you look at the position after 27.Ng3 it looks pretty depressing for white. Black has all the play, d3 is weak and white is lacking space. But 27…c4 gave Adams the possibility to create counter play against the black king, by sacrificing his knight on h6.

Concerning wisdom: I'm not sure you will have much luck searching for heuristics. Counter play and prophylaxis mostly rely on accurate calculation and maybe some alertness for tactical possibilities. Apart from that it is just an aspect of your positional feel for the game, to know how much play you can allow your opponent and when you have to throw a spanner into his works no matter what the cost.