means A situation, especially in negotiations or conflict, where no party can make progress and no resolution is possible — borrowed from the chess position where a player who cannot move legally yet isn't in check, forcing a draw.

from From chess. The second half is the familiar 'mate' of 'checkmate,' which comes through Old French from the Arabic-Persian phrase 'shah mat,' often glossed as 'the king is helpless' (the popular 'the king is dead' reading is likely a later embellishment). The first half, 'stale-,' is murkier: probably from an old word meaning 'fixed in place' or 'standing still,' a cousin of 'stall' — the king stuck, unable to move yet unconquered. The figurative sense of a deadlock between rival parties came later by obvious analogy.