Chaturanga is an ancient Indian game which is presumed to be the common ancestor of chess, xiangqi, shogi, janggi and makruk. It has been played since at least the seventh century A.D., and is popularly believed to be the oldest chess-like game. Chaturanga is the direct ancestor of shatranj which was the form that brought chess to medieval Europe.
 

Rules
Chaturanga was played on a 8x8 uncheckered board, called Ashtāpada. The board had some special markers, the meaning of which is unknown today. These markers were not related to chaturanga, but were drawn on the board only by tradition. Some historians have conjectured that the Ashtāpada was also used for some old race-type dice game where these markers had a meaning.


Ashtapada. Chaturanga was played on uncheckered 8x8 board, sometimes with special markers. 
The exact rules of the game are not known, however chess historians believe that the game had the same or very similar rules to Shatranj. There is however uncertainty about move of the elephant, the precursor of the Bishops in modern chess. Three different moves are described in ancient literature:

Two squares in any diagonal direction, jumping over one square, as the Alfil in Shatranj. The same move is used for the Boat in a four-handed version of Chaturanga, Chaturaji. This is likeliest its oldest move. The Elephant in Xiangqi (Chinese chess) has the similar move, but without jumping. 
One square forward or one square in any diagonal direction (think of the four legs and trunk of the elephant). This is the same move as the Silver General in Shogi. In Makruk (Thai chess) and Sittuyin (Burmese chess) the elephant moves in the same way. This move was described ca. 1030 by Biruni in his India book. 
In any orthogonal direction jumping over one square. A piece with such a move is called a Dabbābah in some chess variants. This move was described by the Arabic chess master al-Adli ca. 840 in his (partly lost) chess work. (The Arabic word dabbābah in former times meant a covered siege engine for attacking walled fortifications, and nowadays means "army tank"). 
Al-Adli also mentions two further difference to Shatranj:

Stalemate was a win for a stalemated player. This rule seems to be quite illogical, however it appeared again in some medieval chess variations in England ca. 1600. 
The player, who is first to bare the opponent's king (captures all the pieces except the king), wins. In Shatranj this is also a win, but only in the case that the opponent cannot bare the player's king on the next move in return. 

History
"Chaturanga sena" stands for fourfold military: Infantry, Cavalry, Archers and the elephant riders. In Sanskrit, "Chaturanga" literally means "having four limbs or parts" and in epic poetry often means "army". The game was also known under the name "BuddhiBala" or literally "power of the brain".

Chaturanga is based on the fivefold divisions of the ancient Indian army:-

Infantry represented by a line of advancing pawns. 
The King and his advisor or general in the center. 
Thundering war elephants near the center of the army. Later, this rather weak piece was thought not to be a suitable representation for the power of the real elephant in war in India; this caused a change of move and of name, and often in India nowadays the Rook is called the Elephant and the Bishop is called the Camel. 
Mounted cavalry represented by the horse with a move that facilitated flanking. This became the Knight with the distinctive move that marks a game as a likely descendant of Chaturanga. 
Chariots on the wings which move quickly but linearly and became the Rook in Europe, but a ship as chess moved north into Russia. 

See also
Origin of chess 
Chaturaji, four-handed version of Chaturanga 

References
A History of Chess, H.J.R. Murray (1913), ISBN 0936317019. 
The Encyclopedia of Chess Variants, D.B. Pritchard (1994), ISBN 0952414201.