Fundamentals

CAPABLANCA advised that the best way to improve at chess is to study endings.
He explained that endings can be mastered by themselves, whereas openings and the middlegame have to be studied in relation to the endgame.
The simplest of all endings, at least from an abstract view, are pawn endings. And yet in practice they often feature some of a game's worst mistakes.
What follows are what I hope can be justly described as instructive pawn endings, starting with some of my earliest competitive games.

Black has just captured on e5 in Spanton (140) - G Hanson (120), Doncaster & District Division One 1980

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Material is equal. Black has more space in the centre, while White has fewer pawn-islands, but neither factor is enough to upset the equilibrium.
However, the non-symmetrical pawn structures mean the ending is far from trivial.
25.Ke2 Ke7
Black offered a draw.
26.Kd3 d5 27.c4!?

How should Black proceed?

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27...d4?
Creating a protected passed pawn is often a winning plan, or at least one that ensures defeat is out of the question. But as usual with chess, everything depends on the position. Here it is almost certainly a losing move.
Correct is 27...c6, when 28.cxd5, in order to create a passed pawn on the queenside, is only a draw as Black will get a protected passed pawn after 28...cxd5.
Also an improvement on the text is 27...Ke6?!, although after 28.cxd5+ Kxd5 Black's pawn weaknesses mean White has the better game.
28.b4?
It is better to immediately attack the black centre with 28.f4.
If 28...exf4, then 29.Kxd4 is very strong for White thanks to his advanced king, eg 29...Kd6 30.h4 g6 (30...h6? 31.h5) 31.b4 h6 32.Ke4 g5 33.hxg5 hxg5 34.Kf5 etc.
If 28...Ke6, then 29.fxe5 Kxe5 30.b4, and Black will eventually be in zugzwang and have to give up the d pawn.
28...Kd6 29.a4??
This turns a win into a loss. Still correct was f4.
29...a5! 30.bxa5
Or 30.b5 Kc5.
30...Kc6?
Winning is 30...c5, eg 31.f3 Kc6 32.h4 Kb7 33.g4 Ka6 34.Kc2 Kxa5 35.Kb3 d3! 36.Kc3 Kxa4 37.Kd3 Kb3, and this time White will end in zugzwang and so lose the c4 pawn.
31.f4 exf4 32.Kxd4

Black to play and draw

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32...g5
Stockfish10 and Komodo10 point out Black draws with 32...Kb7! One line runs 33.Ke5 Ka6 34.Kxf4 Kxa5 35.Ke4 Kxa4 36.Kd5 Kb4 37.h4 h5 38.Kc6 (not c5??, either here or on the previous move, as Black replies ...Kb5 and White will eventually lose his c pawn to zugzwang) Kxc4 39.Kxc7, and the game is drawn.
33.c5?
Winning is 33.Ke5 (or 33.Ke4) as White cleans up on the kingside (Black cannot ever capture the c4 pawn as the white a6 pawn queens).
33...g4?
Still drawing is ...Kb7!
34.a6 h5 35.Ke4 1-0
Dreadful play by both sides, but all of the bad moves were plausible at first sight. However, first sight in pawn endings is often not enough as they can be much deeper than they appear.