1. A tempo is a move, more or less -- the turn used to make a move. You can gain or lose (or save) a tempo by your choices, especially in the opening.
2. In the Modern Attack against the Two Knights' Defence, White paradoxically can gain a tempo by playing, not f2-f4, but f2-f3 (Ng5) then f3-f4
3. In the Queen's Gambit, White will delay moving the Bf1, hoping to save a tempo on Bf1xc4 instead of Bd3, dxc4, Bxc4
4. In the Nimzo-Indian, White can force Black to take on c3 -- but maybe Black was going to take anyway without being asked. The possible loss of tempo is balanced by having certainty about the type of position we are going to play.
5. In an endgame, where Zugzwang may lurk, you may want to lose a tempo. (Trebuchet). There is a special technique to do so: triangulation. (Alburt-Kasparov)

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[Event "?"]
[Site "?"]
[Date "????.??.??"]
[Round "?"]
[White "Tempo Struggle in the QGD"]
[Black "?"]
[Result "*"]
[PlyCount "15"]

1. d4 d5 2. c4 e6 3. Nc3 Nf6 4. Bg5 Be7 5. e3 O-O 6. Nf3 Nbd7 7. Bd3 (7. Rc1 $14) 7... dxc4 8. Bxc4 $11 *

[Event "?"]
[Site "?"]
[Date "????.??.??"]
[Round "?"]
[White "Tempo gain in the Two Knights'"]
[Black "?"]
[Result "*"]
[PlyCount "22"]

1. e4 e5 2. Nf3 Nc6 3. Bc4 Nf6 4. d4 exd4 5. e5 Ne4 6. O-O d5 7. Bb5 Bd7 8. Bxc6 bxc6 9. Nxd4 Be7 10. f3 (10. f4) 10... Ng5 11. f4 Ne4 *

[Event "?"]
[Site "?"]
[Date "????.??.??"]
[Round "?"]
[White "A tempo to clarify the position"]
[Black "?"]
[Result "*"]
[PlyCount "16"]

1. d4 Nf6 2. c4 e6 3. Nc3 Bb4 4. a3 (4. e3 c5 (4... O-O) (4... b6) (4... d5) (4... Nc6) 5. Bd3 Nc6 6. Nf3 Bxc3+ (6... d5) (6... O-O) 7. bxc3 d6 8. O-O e5) 4... Bxc3+ 5. bxc3 c5 6. e3 Nc6 7. Nf3 d6 8. Bd3 e5 *

[Event "Trebuchet"]
[Site "?"]
[Date "????.??.??"]
[Round "?"]
[White "?"]
[Black "?"]
[Result "*"]
[SetUp "1"]
[FEN "8/8/1k1p1K2/3P4/8/8/8/8 w - - 0 1"]
[PlyCount "3"]

{[#]} 1. Ke7 (1. Ke6 Kc5 {Black wins}) 1... Kc5 2. Ke6 {White wins}*

[Event "46th URS-ch selection"]
[Site "Daugavpils URS"]
[Date "1978.07.??"]
[Round "8"]
[White "Lev Alburt"]
[Black "Garry Kasparov"]
[Result "0-1"]
[ECO "E75"]
[PlyCount "112"]
[EventDate "1978.??.??"]

1. c4 Nf6 2. Nc3 g6 3. d4 Bg7 4. e4 d6 5. Be2 O-O 6. Bg5 c5 7. d5 h6 8. Bf4 e6 9. dxe6 Bxe6 10. Bxd6 Re8 11. Nf3 Nc6 12. O-O Qa5 13. Nd2 Red8 14. Nb3 Qb6 15. Na4 Qb4 16. Nbxc5 Bxc4 17. Bxc4 Qxc4 18. Nxb7 Nxe4 19. Rc1 Qb5 20. Nxd8 Rxd8 21. Qc2 Nd4 22. Qxe4 Ne2+ 23. Kh1 Nxc1 24. Rxc1 Rxd6 25. Qc2 Qg5 26. Rd1 Qf5 27. Qc1 Rxd1+ 28. Qxd1 Qxf2 29. Qg1 Qc2 30. b3 Qxa2 31. Nc5 Qd2 32. Qb1 Bd4 33. Nd3 Qe3 34. Nb4 h5 35. Qd1 h4 36. Nc2 Qxb3 37. Qxd4 Qb1+ 38. Qg1 Qxc2 39. Qxa7 Qd1+ 40. Qg1 Qxg1+ 41. Kxg1 Kg7 42. Kf2 Kf6 43. Ke3 Ke5 44. Kf3 f5 45. Ke3 g5 46. h3 Kd5 47. Kd3 Kc5 48. Kc3 g4 49. Kd3 gxh3 50. gxh3 Kd5 51. Ke3 Ke5 52. Kf3 f4 53. Kf2 Ke4 54. Ke2 f3+ 55. Kf1 { Black now loses a move by triangulating with the King: Ke4-f5-e5-e4} Kf5 $1 (55... Ke3 56. Ke1 f2+ $4 (56... Kf4 57. Kf1 Ke4 58. Kf2 Kf4 59. Kf1 Kg3 $19) (56... Ke4 57. Kf1 Kf4 $4 (57... Ke5 58. Ke1 Kf5 59. Kf1 Ke4 60. Kf2 Kf4 $19) 58. Kf2 $11) 57. Kf1 Kf3 $11) (55... Kf4 56. Kf2 Ke4 57. Kf1) 56. Kg1 Ke5 $1 (56... Ke5 57. Kf1 Ke4 58. Kf2 Kf4 59. Kf1 Kg3 $19) 0-1