The Weakest Links were the eponymous losers of The Weakest Link, as opposed to the Strongest Link. Typically, a Weakest Link was chosen by a vote performed by the contestants at the end of each round. As votes were made strategically, contestants who performed worst in a round, making them technically the Weakest Link, could be spared by the vote.
Aboard the Game Station, the Weakest Links were ostensibly killed by a disintegrator beam fired from the Anne Droid; any contestant who attempted to escape was automatically branded a "Weakest Link" and disintegrated as well. Just prior to eliminating a contestant, the Anne Droid would routinely announce "You are the Weakest Link. Goodbye." In reality, the Weakest Links, along with all those who lost aboard the Game Station, were in fact transmatted to the hidden Dalek fleet for processing into Daleks of human origin.
In the last known edition of The Weakest Link, the Weakest Links were, in succession, Fitch, Broff, who was not voted but made an attempt to escape, Colleen, Agorax, and Rose Tyler, with Rodrick emerging as Strongest Link. Though the Ninth Doctor and Jack Harkness stormed the set too late to save Rose, they later rescued her from the Dalek Flagship. (TV: Bad Wolf)
During the Battle of the Game Station, the Anne Droid was repurposed to fight against the invading Daleks. Branding her opponents as the "Weakest Link", she transmatted three Daleks who had asked her to identify herself, but before she could transmat a fourth, the Dalek used its gunstick to destroy her head, taking advantage of the fact that she needed to say, "You are the Weakest Link. Goodbye.", before firing. (TV: The Parting of the Ways)