The most common interpretation is the one held by most contemporary societies. Brain death = death. However, there are minor groups who believe:
1) A person is dead only when they irreversibly stop breathing. This is based on the passage where God first breathes into Adam. They also define life as starting as "at first breath".
2) A person is dead only when their hearts irreversibly stop. This probably based on common sense from most societies back before we could do proper brain scans.
The problem arises because all the good organs are irreparably damaged when oxygen no longer reaches them. This the problem with both minor interpretations above. The controversy is that:
1) There is nothing that would lead to an interpretation that receiving organs from a "murdered" person is wrong. Rejecting the organ doesn't undo the deed and, in any case, there is now a potential for a saved life as a result of the act they believe immoral. So they accept the organs, which makes other
2) People think that they're getting a "free ride". Essentially, while these potential organ-donors are behaving semi-altruistically, they believe those who chose not to be donors are violating a social contract. So while they can't force people to give up their own organs, they are trying to pass legislation where there is now a negative consequence for not participating. This irritates
3) Altruistic donors, who do not care a whit about the argument and just want to save lives. They don't want their organs being used as pawns in a political/religious game when the whole reason they signed the consent form was to help those in the most need. The problem is that those who have the most need would be bumped below those who did not need the organ as much.