The endgame is defined as the stage of the game when few pieces are left on the board. Passed pawns (pawns that are far advanced on the board and/or have a chance of promoting to a queen) are also key elements to the endgame. In addition, the king becomes very important due to its diagonal movement. In summing up the definition of the endgame, Max Euwe and Walter Meiden, prominent chess figures in the mid-20th century, developed five generalizations of the endgame.
1. In king and pawn endings, an extra pawn is decisive in more than 90 percent of the cases.
2. In endgames with pieces and pawns, an extra pawn is a winning advantage in 50 to 60 percent of the cases. It becomes more decisive if the stronger side has a positional advantage
3. The king plays an important role in the endgame.
4. Initiative is more important in the endgame than in other phases of the game. In rook endgames the initiative is usually worth at least a pawn.
5. Two connected passed pawns are very strong. If they reach their sixth rank they are generally as powerful as a rook.
Alright, so we have discussed plenty about the endgame, now let's see it in action.
Above is a common endgame with a single pawn in addition to the kings. As the caption states, if white moves 1. Kb6, the a file is completely protected for the pawn to advance and promote. However, if 1...Kc5, white is forced to play 2. Ka6 Kc6 (2...Kb4, results in 3. a5 and road to promotion).
Starting after 1...Kc5. White is forced to play Ka5, resulting in draw.
Below is a common endgame involving many pawns and few minor or major pieces. The main elements of this match are truly between the king and bishops. The move white makes to win is fantastic, e6! (vacating e5 for his king). White goes on to sac his bishop for the pawn at g6, allowing for his h pawn to promote.
Molnar vs. Nagy, 1966
Among one of the most common endgames is between rooks with the addition of a pawn. In the diagram below, white has the advantage with his pawn at e7, but only if it's his move next. Otherwise, black can check repeatedly forcing the draw. 1. Rg1+ Kg6 (if 1. Rf7+ Kg8!) 2. Kf8 Ra8+ 3. e8Q Rxe8 4. Kxe8 and the game is over in about 15 moves.
There are a plethora of endgame variations, trillions in fact. But it is the idea of how to play the endgame and what to look for that counts. Be careful not to stalemate your opponent and avoid zugzwang as well. Of course, not all endgames can be won, but as I said earlier, none should be lost either. Always aim for the win, and accept nothing less than a draw. Emanuel Lasker, world chess champion for twenty-seven years once said, "When you see a good move, look for a better one." Take heed to Lasker's words and may you play well!
Chess images taken from wikipedia.org and caissa.com