The conversion theatre was the area in the hospital on floor 1056 of the colony ship that people were brought to in order to be converted in to patients and eventually Cybermen.
It was ran by the surgeon assisted by the nurse. Just past the theatre was the lifts to get to the other floors, such as Floor 0000 or Floor 0507. The The Master had a key to enter it, luring Bill in side to be converted. The Twelfth Doctor, Nardole and Missy entered the area, finding the cyber converted Bill. (TV: World Enough and Time)
It housed the computers to the Cyber-net, the Doctor using them to define humans as also having two hearts so they would attack the Master as well. (TV: The Doctor Falls)