Like the standard game of chess, but the pieces behind the pawns on the white side are arranged at random, with the proviso that bishops must end up on opposite colors, and the king is somewhere between the two rooks. The black pieces are lined up to mirror the white.
This results in 960 different possible positions, hence the name Chess960:
4 possible white bishop locations
× 4 possible black bishop locations
× 6 possible queen locations
× 5 × 4 / 2 = 10 possible knight arrangements
= 960 possible starting setups
The game was proposed by Grand Master and former World Champion Bobby Fischer, to remove the need to memorize openings. He originally called it Fischerandom, and later changed the name to Fischer Random Chess, which is also frequently used. The name Chess960 was proposed by a FIDE committee and is now the official one.