Is being brain dead equivalent to being in a coma?
No, actually, but people often get the two confused. The definition of brain death is that the entire brain has irreversibly lost all function. Everyone who dies, ultimately dies of brain death even if they didn't suffer a direct injury to the brain. For example, when people die from heart attacks, the real problem is that their brain isn't getting oxygen, which is why knowing CPR can make all the difference.
Comatose But Still Firing
Some people in comas need life support to breathe and regulate their heart rate and blood pressure because their brains have stopped doing their jobs and are irreversibly damaged. But people who are in a coma haven't lost all brain function. That is, they still have some level of measurable brain activity and their brains are still sending out electrical impulses in response to stimulus.
For example, people in a coma may still have a gag reflex, but people suffering from brain death don't. And even those people who are in a really deep coma have enough brain voltage to show up on an EEG, a test so sensitive it can pick up the static electricity on clothes.
"How Brain Death Works" (How Stuff Works)