If I am shooting the eight-ball for the game, hit the eight-ball, but sink the opponent's ball and the eight-ball does not go down, what happens?
It is your opponent's turn.
You were on the 8 and designated the pocket. With the cue ball, you struck your ball first, the 8 ball, which is the 1st requirement for a legal hit. A ball - as in any ball - was pocketed, or hit a rail, which meets the 2nd and final requirement for a legal hit. There was no foul, nor did you pocket your ball. Therefore, your turn is over.