Brain-damaged means a person’s brain has been hurt — by a crash, a fall, a stroke, a high fever, or a time with too little air — so that thinking, moving, speaking, or remembering may not work as they did before. It is a medical adjective, and it is heavy. How much trouble a person has depends on which part of the brain was hurt, and how badly; two people with the same word on a chart may live very different lives. Doctors still say it, though actually many careful speakers now prefer a person with a brain injury, because the old adjective can sound as if the person is only their damage. People stretch the word. Online, some speakers use it as a cruel joke for “stupid,” close to Taiwanese 腦殘. That is loose, and it can hurt classmates who live with a real injury. In old computer talk, engineers also called a badly built system brain-damaged: that joke grew from a made-up “disease” they named Honeywell Brain Damage. Keep the medical sense in mind, and be slow to put this word on a person unless you know what you mean. We usually write the hyphen when it sits before a noun: a brain-damaged patient.