Tactics
The zwischenzug
In the middle of a sequence that looks forced, you play something else first. A check, a bigger threat, a capture they have to answer. Then you go back and finish the sequence, having gained something for free. Also called an intermezzo, and it is the reason a calculation that ends 'and then I recapture' is not finished.
How to spot it
Every time a recapture feels automatic, stop and ask what else is forcing. The test is whether the recapture will still be available after the in-between move, and the usual reason it will is that the piece you would recapture cannot run anywhere useful while it is your opponent's turn to deal with a check.
The positions
Easiest first. Every one is from a real game, and the rating under each board is what it costs the players who have tried it on lichess.
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1. White to play Play it out →Answer
28.Re8+ Kh7 29.gxf3
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2. White to play Play it out →Answer
13.Nxf6+ Bxf6 14.Qxe4
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3. Black to play Play it out →Answer
24...Qxe3+ 25.Kc2 Bxe4
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4. White to play Play it out →Answer
38.Rf5+ Kg6 39.Rxd4
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5. White to play Play it out →Answer
36.Rxe6+ Kb7 37.Qxf7+ Qc7 38.Re7 Qxe7 39.Qxe7+
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6. Black to play Play it out →Answer
14...Rxf2+ 15.Rxf2 Qg3+ 16.Kh1 Nxf2#
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7. White to play Play it out →Answer
9.Nxd6+ Qxd6 10.fxe5
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8. Black to play Play it out →Answer
40...cxb2 41.Rb8 Nxa4
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9. White to play Play it out →Answer
25.Rg3 g6 26.Rxc5
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10. Black to play Play it out →Answer
16...Bxb5 17.Nxb5 Nb2 18.Qb3 Rxb5
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11. Black to play Play it out →Answer
36...g2+ 37.Rxg2 Rxh2+ 38.Rxh2 Rg1#
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12. White to play Play it out →Answer
11.Bxh7+ Kxh7 12.hxg5+ Kg8 13.Rh8+ Kxh8 14.Qh5+ Kg8 15.g6 Nf6 16.exf6 Rxf6 17.Qh7+ Kf8 18.Qh8#
Reading about a pattern is the easy half. Motify finds the positions where you missed one, by running your own lichess and Chess.com games through Stockfish, and puts each of them behind two or three puzzles on the same idea so you meet it having just found it three times. See how Motify works.
Positions from the lichess puzzle database, released under CC0.