If someone wishes to join your club and you blackball him, what you are doing is voting against him. You are informing the members of your club that you do not wish this particular individual to become a member of your club.
The expression goes back to a practice that was followed in ancient Greece. Admission to a club/society was determined by all members of the club/society. All members had to vote; they had to indicate whether the new individual should be let in or not. The voting was done in secret. Each member had to drop a pebble in the shape of a ball into a box. The pebbles were usually of three colours — red, white and black. If all the pebbles that were put in the box were either white or red, then the candidate was admitted. But if there was even one black pebble, then the candidate was not let in.