大脑(dà nǎo) - brain, cerebrum in Mandarin

大脑 (dà nǎo) is the brain, and strictly the cerebrum, as opposed to 小脑 (xiǎo nǎo), the cerebellum. In ordinary speech it just means the brain as the organ you think with.

There are three words in this space and they are not interchangeable. 大脑 is the anatomical and semi-technical term, the one a doctor or a science article uses. 脑子 (nǎo zi) is the colloquial word for one's brains or wits: 动动脑子 means use your head, and 脑子进水了 is a rude way of saying someone is not thinking straight. 脑袋 (nǎo dai) means the head as a physical lump on your shoulders.

The trap for English speakers is that "brain" in the figurative sense of intelligence usually maps to 脑子, not 大脑. 他很有头脑 or 他脑子很好 means he is bright; 他大脑很好 sounds like a medical report.