Tactics
The knight fork
Every piece can fork, and the knight is the one that hurts. A rook forking two pieces can be blocked or chased; a knight attacks in a shape no other piece moves in, so it cannot be blocked at all, and the only answers are to take it, move both targets, or defend both.
How to spot it
Knights fork on squares of one colour, so scan the squares of the colour your knight is not standing on and ask which of them touches two enemy pieces. The shape to memorise is two enemy pieces two files and one rank apart, or one file and two ranks: that is a forking square waiting for a knight.
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. Black to play Play it out →Answer
10...Ng2+ 11.Kd2 Nxe3
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2. Black to play Play it out →Answer
32...Ng3+ 33.Kg2 Nxf1
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3. White to play Play it out →Answer
28.Ne7+ Kh7 29.Rh2#
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4. White to play Play it out →Answer
13.Nd6+ Kd8 14.Nxe4
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5. Black to play Play it out →Answer
22...Nxf3+ 23.Kg2 Nxd2 24.Bxb6 Nxe4
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6. Black to play Play it out →Answer
22...Nc2 23.Rc1 Nxe3
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7. Black to play Play it out →Answer
29...Nxf4 30.Qg5 Nxd3+
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8. White to play Play it out →Answer
23.Ne7+ Kh8 24.Qxf8#
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9. Black to play Play it out →Answer
28...Nb8 29.Qe8+ Qxe8
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10. Black to play Play it out →Answer
25...Nd5 26.exd5 Rxe3
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11. White to play Play it out →Answer
27.Ne7+ Kh8 28.Rxh7+ Kxh7 29.Qh4+ Qh5 30.Qxh5#
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12. White to play Play it out →Answer
40.Ne7+ Kg7 41.Nxg6+ Kxg6 42.Qxg3+
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.