BASIC CHESS ENDGAMES
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ROOK ENDGAMES
In Rook endgames counterattack is the best defense, as the Rook is a very strong piece if employed to attack while it's very weak as a defender.  Those endgames are very complex and require a precise play.  Material advantage can't always guarantee a win.
ROOK 1 ROOK 2
ROOK 3 ROOK 4





PAWN ENDGAMES
Gaining the opposition is extremely important in pawn endgames.  Opposition occurs when the two kings face each other on a rank or file, with only one square in between them. In such a situation, the player that doesn't have to move achieves the opposition.  The two Kings can face each other along a column (vertical opposition) or a rank (horizontal opposition) or on the same diagonal (diagonal opposition). In all these case we still have opposition when an odd number of squares are between the two Kings.
PAWN 1 PAWN 2
PAWN 3 PAWN 4

BISHOP vs KNIGHT ENDGAMES
It is not possible to state whether the Bishop or the Knight is stronger in chess endgames: sometimes the Bishop is stronger, sometimes the Knight.  As a general rule the Bishop can contrast better passed pawns, can control distant squares but can only reach half of the squares; on the other side the Knight can jump pawn chains and can reach any square.  In the below endgames the Knight is stronger  in BK_1 and BK_2 while the Bishop in BK_3 and BK_4.
BK 1 BK 2
BK 3 BK 4


OPPOSITE COLOUR BISHOPS ENDGAMES 
Opposite colour Bishops endgames are well known for their drawish character.  That often it's true, but some cases exist in which the player with material advantage can force the win all the same.   Therefore it's important to familiarize with some typical endgames K+B+2P vs K+B that can be classified according the pawn structure (doubled pawns, isolated pawns, adjacent pawns).
OPPOSITE BISHOPS 1 OPPOSITE BISHOPS 2
OPPOSITE BISHOPS 3 OPPOSITE BISHOPS 4

The following endgame is taken from a real game (60th Italian Correspondence Chess Championship - Semifinal A) and it is an
in which it is possible to force a win.

SAME COLOUR BISHOP ENDGAMES
In the endgames with bishops of the same colour it's important to be capable to assess the simple ending K+B+P vs K+B. As a general rule this ending is draw if the defending King is able to occupy any square in front of the pawn, provided that such square is opposite in colour to the squares the Bihops travel on. Instead if the defending King is behind the pawn and the attacking King is near the pawn  the outcome of the ending is defined by Centurini rule.  
SAME BISHOP 1 SAME BISHOP 2
SAME BISHOP 3 SAME BISHOP 4