I think they're KO'ed (or they faint) in most of the FFs. Characters with 0 HP died in the first few FFs, however, but back then, it was very hard to revive them. For example, in the original FF1 for NES, you could only revive people on clinics/churches or by using Life spells outside battles.

In FFV, it's implied that the characters reaching 0 HP doesn't mean death. When (SPOILER)Galuf dies, the party members try to use Cure spells, Life spells and a Phoenix Down on him, but none of them work anymore. In FFT, the characters faint, but if nobody revives them in 3 turns, they die, turning into a crystal and being gone for good.