In Chess what is the Queens or Kings Indian?
At a high level, an "Indian" in Chess is just advancing the pawn which is in front of a knight by one square.  

King or Queen indicates which side the advanced pawn is on.  This move will usually be followed by the fianchetto of the bishop nearest the pawn.  This puts the bishop in front of the knight and allows him to control one of the long diagonals on the board from a safe location.

Two common openings using this are:


The King's Indian Attack, usually done in response to the French defense.  A famous example is from 1967.  Bobby Fisher against Lhamsuren Myagmarsuren.
The Queen's Indian Defense starts like this 1.d4 Nf6 2.c4 e6 3.Nf3 b6