The Clockwerk Brain is the Clockwerk Parts of Clockwerk that possesses his mind and being. Without this part, Clockwerk would not be able to reactivate himself.
Clockwerk replaced his organic brain with a metal one so he could not age and also become immortal.
Sly Cooper and the Thievius Raccoonus
The brain could briefly be seen in Sly Cooper and the Thievius Raccoonus when Sly was destroying Clockwerk's head.
Sly 2: Band of Thieves
The brain is first mentioned in Sly 2: Band of Thieves, first in "He Who Tames the Iron Horse." In "Cabin Crimes," one of the cabins Sly Cooper enters hosts Jean Bison, who, at the time, is having a conversation with Arpeggio. A short way into the conversation, Bison mentions his desire to buy the brain off of Arpeggio, but Arpeggio mentions that he already has the lion's share of the parts, which consists of the Clockwerk Lungs and Stomach.
Another mention is in "Menace from the North, eh!." When Sly exits the safe house, Bentley talks to him through his binocucom, informing him that if they are ever going to take a peek at the brain and steal it from Arpeggio, they will need to sneak aboard his blimp