It is very much important to have passed pawn in endgame. Sometimes material on the board is even but it may be possible to have passed pawns for both on different wings (King side, Queen side. In endgame it is advantageous to have passed pawn on the side where your opponent has not castled. In general that is called majority on the queen-side. It is advantageous because your opponent king has to travel far to reach your passed pawn by the time you can just take off some of his pawns and gaining material advantages.
Pawn Breakthrough :
It is basically a process of creating passed pawn.In simple terminology; finding correct series of pawn moves to produce passed pawn called pawn breakthrough.First we will go through some famous examples then we will dig this topic deeply.
Remember : You must be having prior knowledge of 'key square' and 'Rule of square' in order to understand pawn breakthrough.
Position 1: White to Move
This is very famous example to explain pawn breakthrough. See the diagram(^). White white to move, white can easily win with
1. g6!! hxg6 or 1....fxg6, 2.h6! gxh6, 3.f6 and wins
2. f6! gxf6
3. h6 and promoting the pawn and wins.
But there is question why white was winning on the given position?
- Black king was too much far from the queening square.
- White pawns were more advanced.White won't be able to win if pawn were on the 4th rank and black's pawn on 6th rank.
Position 2: White to Move
In this position(^), white can win with either with f5 or g5
1. g5 Kb4
2. f5 (threat is to g6 on next move, creating passed pawn on e file)exf5
3. g6!! fxg6
4. e6 and black king is not far from queening square.
again winning is thanks to white advanced pawn and king's positions.
In the both diagrams, kings were far from the action. But it is very much harder to evaluate position when king is in action. Let's look at some examples.
Position 3: White to Move
In the diagram(^), if black king was far from the action then white can easily win with 1. b6 - axb6 2. a6 but here position is different, as if 2. a6 then Kc6 and black can win even therefore much deeper calculation is required
1. a6! Kd6
2. b6!! Kc6
then pause for moment which one is better b7 or bxa7, yeap may be you got right
3. axb7! and wins (as b7 draws because 3...Kc7 and black can save a day due to stalemate tricks)
Position 4: Black to Move
In the given position(^) with white to move white can win easily Kd4, but if black to move then it is his or her only chance, How he/she can win? with 1....c3 2. bxc3 bxc3 3.Kd3 and white can easily get the black's passed pawn.
Think for some more time, may be you get that
2. bxc3 b3!!
3. axb3 a3 which is not in the reach of white's king.
So there is no rocket science to learn pawn breakthrough, but below are some examples in the form of puzzles which you should try to solve first then look at the answers.
(1) White to Move
(2) White to Move
(3) White to Move