The long string is an imaginary line from the center diamond on the head rail to the center diamond on the foot rail. It splits the table in half longways. Basically, it's a foul for your ball to cross over into the other shooter's side of the table.
It's not really a foul (and you probably didn't mean it that way), it's a "bad" lag. If the opponent's lag is not bad, the opponent wins the lag. If the opponent's lag is also bad, the players lag again.