The bowls first tether the user to a person or several different people and then the Philosopher's Stone drains those people of their collective life energy, and transfers it to the user making the user immortal. However, any person that was tethered to the user by the bowls will die instantly as soon as the user becomes immortal. Also, for them to become immortal, the user must be near a lot of petrified wood from the Euphrates River region.
It's unknown exactly how many people must be sacrificed to make the user immortal, but in order to make Nick, Sutton, and Charlotte immortal Paracelsus had to sacrifice the lives of 600 people and to make himself immortal he sacrificed 200 people.
To tether someone, the Philosopher's Stone must be placed inside the bowl bearing the ankh symbol; to immortalize someone, the Stone must then be placed into the counterpart bowl and be heated there to initiate the process.
In the episode "The Truth Hurts " when Paracelsus was using the bowls, he specifically targeted terminally ill patients in hospitals, because their weakened immune systems made it easier for him to leach off their life energy.