脑子(nǎo zi) - brain, mind, head in Mandarin

脑子 (nǎo zi) is the everyday word for the mind: your wits, your memory, the thing you think with. 脑子快 is quick-witted, 脑子里 is inside your head, and 动动脑子 is have a think.

For the organ itself, a doctor says 大脑 or 脑部. 脑子 belongs to conversation, and much of the time it is a rough compliment or a rough insult: 有脑子 means someone has their wits about them, 没脑子 that they have not got any.

The trap for English speakers is treating it as a straight swap for "brain". You would not say "my brain is not clear today" in English, but 我脑子不清楚 is completely normal Chinese. Conversely, "a brain scan" is 脑部检查, never 脑子检查. Think of 脑子 as "head" in the sense of "use your head" and you will place it correctly.