Each piece can make two clones of itself, taking a whole turn. Clones must be placed on the adjacent squares (on the same vertical, horizontal or diagonal. The only exception is Knight, but more on this later, as the explanation would probably require to see this on a real board*) . If the adjacent squares are occupied, the move is considered illegal. Then player can choose which cloned piece he considers as "real" (let's call other clones "fakes"). If the "fake" piece gets attacked, it disappears by the next move . Similarly, if the "fake" King gets checked it also disappears. The "real" King might ignore checks, but if it gets captured, the side that captured King, wins. Each piece can make clones only once per the game. Pawns can't make clones.
* about Knights. Knight clones must be placed on the free squares that Knight can move to:
To avoid too much compications, clones of a Knight must be placed on the squares of the "same direction": if a Knight on g1 decides to place a clone on f3, the second clone must be placed on e5 . Second example: Black is threatening mate with the Bishop+Queen battery, but White replies with Nd2@c4@b6+ and wins Black's Queen.
This makes a Knight a long-range, and probably one of the most dangerous piece ever (after the Queen, probably).
UPD. Similarly, Bishops, can make only make clones along the same diagonals they move, the same rule applies to Rooks (they only make clones along the vertical or horizontal they move).
UPD2. Fake clones of the piece should be indistinguishable from the real piece for the opponent (at the same time, player should be able to distinguish fake clones of own pieces from the real piece).