All king and pawn endgames usually come down to who can queen first, and usually the side with more pawns is the only side that can queen at all. Of course there are exceptions:
8/8/3P4/2k3p1/2P5/Pp5p/1P5K/8 b
Black to Move and Win
Even though Black has fewer pawns, he can queen first. Black begins with 1. ... g4. Now, if White plays 2. d7 then he will get steamrolled with 2. ... g3+ 3. Kg1 h2+ 4. Kf1 (4. Kh1 g2#) h1=Q+ 5. Ke2 Qf3+ etc. [0-1]. Instead, White can try to connect kings: 2. Kg3 h2 3. Kf4 h1=Q 4. Ke5 Qd5+ 5. Kd4 Kb6+ [0-1]. Now White is lost because 6. Kc5 runs into 6. ... Qxd6##.
So, besides having more pawns, other factors that can determine who will win a king and pawn endgame include who has the more advanced passed pawns and who has the better placed king (White's king in the above diagram is in a bad spot because the Black pawns can win a few tempos by checking him). Most endings like the above can be analyzed quickly on the spot. All that is usually required is the counting of tempos.
However, a more nontrivial unequal pawn ending is one where the superior side has one more pawn and can queen first, but still can't win! This can happen if the weaker side can trade all of his own pawns and connect kings. Such an ending would be drawn because king and queen can only defeat a King if the kings aren't connected. Here's an example:
6k1/p5p1/3p1p2/1P3P2/Q3P3/8/2r5/3r1K2 w
White to Move and Draw
White has only one legal move here, but luckily it is a good one: 1. Qxc2. Now, Black has an extra pawn, but cannot win. White's plan is to trade off his b-pawn quickly and then get his king closer to Black's. Play continues: 1. ... d5 2. exd5 g5 3. fxg6 e.p. f5 4. b6 axb6. Black was forced to take the b-pawn that was threatening to queen.
6k1/8/8/5p2/8/8/8/5K2 w
White to Move and Draw
Now, White's task is easy. He will connect kings before Black can queen and mate him. There is plenty of time: 5. Kf2 f4 6. Kf3 Kh8 7. Kg4 f3 8. Kh5 f2 9. Kh6 f1=Q 10. Kh7 draw.
When an ending like the one above is reached, it is usually quite easy to assess whether the draw is possible. The weaker side just needs to decide how it will trade off all its pawns and whether there is time to connect kings. Consider:
8/pp1q4/6k1/3R4/8/8/P3K3/8 w
White to Move and Draw
Here, the question is should White take the queen or check the king? The answer is easy: take the queen! White certainly has no win and taking the queen is an easy draw, so it should be done on reflex. The reason why the resulting position will be drawn is because Black cannot queen without first trading a pawn for White's a-pawn, and White has plenty of time to connect kings. Of course, endlessly checking the Black king is fine too, but why prolong the draw? Note: if White does decide to check the Black king, he shouldn't allow the kings to connect while the heavy pieces are still on the board.
Two-on-one pawn situations like that above are drawn, but three-on-two positions are an entirely different story! Consider:
8/ppp2k2/8/3pr3/1P2P3/8/PK6/8 w
Poroshat vs. mrundersun
White to Move
In this position, White resigned! I consider this the best resignation ever since, as you will see, it takes quite a bit of analysis to realize that Black has a won position.
First of all, it is clear White should play: 1. exd5.
8/ppp2k2/8/8/1P6/8/PK6/8 b
Black to Move
If Black is going to win this position, then he needs to queen a pawn while leaving a White pawn behind. Before we show the winning technique, let's looks at a few attempts that fail. For example, 1. ... a5? 2. b5!
8/1pp2k2/8/pP6/8/8/PK6/8 b
Black to Move
This draws because now White will take the Black c-pawn when it advances and then later capture the b-pawn when it advances (and connect kings in the meantime). Back to:
8/ppp2k2/8/8/1P6/8/PK6/8 b
Black to Move
Another try: 1. ... b5 2. Kc2 a5 3. a4!
8/2p2k2/8/pp6/PP6/8/2K5/8 b
Black to Move
Now, if Black plays bxa4, then White responds with bxa5, and if Black plays axb4, then White responds with axb5. So, Black tries: 3. ... c5 4. axb5! a4 5. bxc5 a3 6. Kd3 a2 7. Ke4 a1=Q 8. Kf5.
8/5k2/8/5K2/8/8/8/q7 b
Black to Move
White connects just in time. Let's go back to the beginning now and show how Black can win:
8/ppp2k2/8/8/1P6/8/PK6/8 b
Black to Move
First of all, White's goal is to trade his b-pawn for Black's c-pawn and his a-pawn for Black's b-pawn. There are two ways that Black can try to stop this: (1) He can trade his a-pawn for White's b-pawn, gaining a passed c-pawn, and (2) He can somehow get his c-pawn past White's b-pawn without trading them. Note that if (1) occurs, then White still might draw, depending mainly on the placement of Black's king. If the king is too far away, then after Black trades his a-pawn for White's b-pawn, White will push his a-pawn in a hurry and Black might be forced to capture it with his b-pawn lest White queen first.
Now, you might think that White can sit tight and not push any pawns and there is nothing Black can do. Afterall, we refuted 1. ... a5 and 1. ... b5 above, and of course 1. ... c5 2. bxc5 leads to a drawn two-on-one pawn ending. However, let's see what Black can really do if White stays put. Imagine White shuffles his king back and forth between b2 and c2, letting Black do whatever he wants with his pieces. Then, Black would form this ideal set-up:
1k6/pp6/2p5/8/1P6/8/PK6/8 w
Black's Ideal Set-Up
Why a pawn on c6 you ask? Well, remember that the reason why a7-a5 failed when the pawn has on c7 was because of the reply b5. But now, Black is genuinely threatening to advance to a5 because then b5 is met by c5! See the difference? No en passant available! Now, in the above position, even if it's White's move, he is lost. A king move loses to a5, moving the White pawn to a4 loses to a5, and moving the White pawn to b5 loses to c5.
So, White must work for the draw if there is one. He can't just wait around. Back to the original position with Black to play:
8/ppp2k2/8/8/1P6/8/PK6/8 b
Black to Move
White's problem right now is that his b-pawn is advanced ahead of his a-pawn. This allows it to be attacked and either traded off or advanced beyond Black's c-pawn. White sould rather have his a-pawn ahead of his b-pawn (that's the key to these three-on-two pawn endings). For example, consider this position with Black to move:
8/ppp2k2/8/8/P7/1P6/1K6/8 b
Black to Move
This position is an easy draw for White. Black's a-pawn can't be traded for White's b-pawn. So, White can safely leave his pawns where they are, and no matter how Black advances his pawns to try and win, White will trade his a-pawn for Black's b-pawn and his b-pawn for one of Black's remaining pawns, and easily connect kings before Black's remaining pawn can queen and checkmate.
In face, from the initial position again:
8/ppp2k2/8/8/1P6/8/PK6/8 b
Black to Move
White would like to cure the affliction of his advanced b-pawn by advancing his own a-pawn to a5. So, if Black sits back and does nothing for a couple of moves, White could achieve his own ideal set-up:
8/ppp2k2/8/P7/1P6/8/1K6/8 b
White's Ideal Set-Up
Again, Black's a-pawn can't be traded for White's b-pawn, so White will leave his pawns on a5 and b4 and easily trade both off if and when Black's pawns advance.
So, Black and White each have their own ideal set-ups to try and achieve, and the truth is that White cannot stop Black from achieving his, but Black can stop White (without messing up his own position).
Now on to the actual analysis. Black can win with different first moves, but let's begin with 1. ... Ke8. Then, here we are:
4k3/ppp5/8/8/1P6/8/PK6/8 w
White to Move
Now White cannot afford to sit around and let Black play c6 and move his king over. He needs to try to get his pawn to a5. Thus: 2. a4. Note that trying to confuse the issue with 2. b5 does no good (not to mention that it is opposed to the general principle of getting the a-pawn ahead of the b-pawn) because of 2. ... a6, and now 3. b6 loses to 3. ... c5, 3. bxa6 loses to 3. ... Kd8 4. a4 Kc8 5. a5 Kb8 6. a6 b5 7. a7+ Ka8 [0-1], and 3. a4 loses to 3. ... Kd8 (not 3. ... axb6?? 4. a5 Kd8 5. a6 bxa6 draw!) 4. a5 Kc8 etc. [0-1].
4k3/ppp5/8/8/PP6/8/1K6/8 b
Black to Move
Ok, now Black must stop White from playing a5 next. The only move that does it is 2. ... b6! (2. ... a5, for example, is met by simply 3. b5 draw).
4k3/p1p5/1p6/8/PP6/8/1K6/8 w
White to Move
Now, 3. a5 is met by 3. ... b5 and Black will trade his c-pawn for White's b-pawn and then queen his b-pawn while the a-pawns remain locked. We will show that 3. b5 loses too, but first, let's see what would happen if White waited around with king moves: 3. Kc2 Kd8 4. Kd3 Kc8 5. Ke4 Kb8 6. Ke5 c6 7. Ke6 a5! [0-1], and Black has achieved his ideal set-up. This leaves: 3. b5.
4k3/p1p5/1p6/1P6/P7/8/1K6/8 b
Black to Move
Now Black must hurry over with his king: 3. ... Kd8!
3k4/p1p5/1p6/1P6/P7/8/1K6/8 w
White to Move
Now, there are two paths (more or less) White could take to his death. First: 4. a5 c5! 5. axb6 (5. bxc6 b5 [0-1]) 5. ... Kc8 [0-1] (now b6 is met with a5 and Black can stop the White pawn with his king). Second: 4. Kc3 Kc8! 5. Kb3 a6! 6. a5 c5! [0-1]
2k5/8/pp6/PPp5/8/1K6/8/8 w
White to Move
Now, if White plays 7. bxc6, then 7. ... b5 wins, if White takes the a-pawn, then Black pushes the b-pawn, and finally if White takes the b-pawn, then Black pushes his a-pawn.
This was a very delicate ending, where the outcome hinged on a mere tempo or two. Move the Black king over to the h-file, say, and White would draw. Anyway, I hope you learned a general principle here in three-on-two pawn endings like this. White wants his a-pawn advanced ahead of his b-pawn (or the equivalent).