Brain death is the legal standard of death, essentially a brain that shows no activity on an EEG. This development was made necessary by the fact that respiration and circulation can be sustained indefinitely through external means.
Many people can suffer severe brain damage without being technically brain dead. For example, a severely brain damaged person may still be able to move, breathe, and maintain digestive function without being able to communicate or respond to stimuli in any way. This is often defined as a persistent vegetative state, and differs from a coma in that coma patients often have no major brain injury whatsoever.
However, a patient may not be brain dead in order to be declared dead. For example, a patient whose heart cannot be restarted will often be declared dead even though heart/lung bypass could keep their brain sustained indefinitely. Death will also be declared when the patient will not respond to stimuli, such as a lack of iris response when a light is shined in their eye.