The Thought Core was the self-aware part of the Godhead. It controlled the power of the Godhead.
The Ministers of Grace separated the Core from the Godhead and destroyed the Godhead, scattering its shards throughout time. Wainwright was assigned to guard the Core.
The Master gathered the shards and reunited them, then searched for the Core. He found it at the Wainwright Museum in Chicago. The Core stole the Master's TARDIS and controlled him and several others in an attempt to be reunited with the Godhead.
The pieces were brought together by Margie del Monsalvat, and the Godhead needed a mind to control it. Margie intended for it to be the Master's mind, but instead it chose the dead body of Father Sherrin. Under his control, it left Earth. (PROSE: The Duke of Dominoes)