Heroes (and their allies, which include animal companions and mercenaries now) die when they are reduced to zero body points (they can't go negative), but in the North American version of the game, they are allowed to save themselves from death under certain circumstances, so being killed isn't necessarily the end of the game for that character. I realize that the way the game is setup isn't to everyone's liking. I don't have a problem with the way the game was setup. So obviously a person could decide to change it to something else at their own table.
Normally a Spell (Spell Scroll is effectively a Spell but in the form of an item that could change hands and doesn't recharge between quests if used) requires an action, but a Potion can be used "anytime" unless the card says otherwise. Artifacts are another story... sometimes they require an action to use, and sometimes not, so you have to read the card. The Elixir of Life is a really cool Artifact, but it is extremely vague on how it is supposed to work, leaving a lot of room for interpretation.
Using the NA rules (here classic, but they don't really change with any of the remake material released so far with one exception I'll relate later) as a base, let's look at the common scenarios:

Hero's turn... Rolls to move, steps on a trap, reducing his body points to zero. He's got a Potion of Healing, the Armband of Healing artifact, a Spell Scroll for Heal Body, and the Water of Healing Spell.*
At this point, any of these can be chosen to automatically kick in and restore him to life. If the thing that brought him back to life was an action then that's his action for the turn. Let's say he doesn't have any of these, then he's dead, and he will remain dead for the rest of the quest, unless someone finds an Elixir of Life that can bring him back at any point. It would be too late by the time another hero had their turn to use a potion or healing spell on him to save him from death.

Hero's turn... He searches for treasure, drawing a Hazard card, that drops his body points to zero. He's got the Spell Scroll and Spell cards, but these can't be used because they require an action, which he's already done. But he could use a Potion or other healing item that didn't require an action.

Hero's turn... He searches for treasure, drawing a Wandering Monster card. The monster does damage in its attack reducing his body points to zero. Same as the above scenario.

Hero's turn... as in the above scenario, but he survives the Wandering Monster's attack. But let's say he is the last Hero in the round order and then it's Zargon's turn. Zargon can use the Wandering Monster to attack AGAIN (the initial attack was part of the Hero's turn) and this time he reduces the Hero's body points to zero with his attack. It's not the Hero's turn, so he can't do any actions, right? Wrong. When it's Zargon's turn there's a special exception that allows the Hero to use an unused healing spell in this scenario to save himself from death IF he has one. Another Hero can't do it for him of course. The same would apply for a Spell scroll since it's treated identically to a Spell. Yes, if he had a healing Potion or other Healing item he could use it here as well.

Zargon's turn... a monster attacks a Hero, reducing the Hero's body points to Zero. That Hero may use any healing item or unused Healing spell he has in that moment to save himself. Despite the healing spell requiring an "action" this is a special exception, the Hero goes to use it even though it's Zargon's turn. Elsewhere in the game there are other exceptions where Zargon gets to do things on the hero's turn as well.

What about unconsciousness? In the expansions, being reduced to zero MIND points has different effects. In Kellar's Keep it's the same as death. Only an Elixir of Life can restore a hero in such a state. Some speculated that anything that could immediately restore your mind points (Potions from the Alchemist Shop in the NA edition for instance) could also be used to save a hero in this scenario. What about after the fact? In Return of the Witch Lord, such a hero reduced to 0 Mind Points is simply unconscious. This implies he is restored to normal so long as at least one other hero makes it out of the quest alive (he dragged his friend's body out too I guess). But gameplay wise an unconscious hero is the same as a dead one, but is his gear lost if a monster stops by (to "loot" the body?). Can someone use a Mind Point restoring potion on him at some later point to restore him mid-quest? You could lose Mind Points in the EU editions as well but recall there were no Alchemist Shops there, so no potions that restored mind points (and no Elixir of Life except that one from Dark Company that brought you back from death with 1 BP only).

In EQP/BQP zero mind points doesn't mean you're dead or unconscious, but rather going into shock with reduced powers. It was never clarified in the classic era whether a Hero could escape the state of shock (before the next quest when everything would be restored to normal it could be assumed) but in the remake era "Into the Northlands" online quest clarified that anything that raises the Hero's Mind Points above zero would cure the state of shock immediately, so this could take place even after the incident from the action or item of another character. Going into shock isn't death obviously, but it's here for the sake of completeness.
* This obviously isn't exhaustive, I'm just including the various things. There are skills (Knight's Stalwart) that can heal too, but Spell Scrolls are treated like spells (costing actions to use) and Potions are free actions. Combat Cards and things like that are homebrew so I didn't really include them. There's also that mulligan spell from the Elf pack that let's you replay a turn that could "undue" a death sequence altogether.

What about Potions, Spell Scrolls and Artifacts? Unlike Spells, these are items that, as a free action, can be handed from one adjacent character to another, right? But remember that a player can only pass an item to another character ON HIS TURN. The receiver is the one who is in trouble so he can't get the item until it's too late! In the EQP/BQP (and in the remake era, in the New Beginnings online quest) the designers introduced the idea that an item can't be passed to an adjacent hero if they are also adjacent to a monster. New Beginnings does NOT specify that it has to be on the turn of the one making the pass. It also doesn't literally say it can only be one item at a time, however Zargon could always choose to interpret it the way he wishes or interpret it strictly based on the earlier rulebook/instructions (no mention of adjacent monsters preventing it but has to be on the turn of the passer) or from the stricter EQP/BQP (has to be on the turn of the passer, adjacent monsters prevent it). Are diagonally adjacent monsters considered "adjacent" ?
It may matter little for most death saves, because most monsters don't have ranged (or diagonal) attacks and so you can expect to be adjacent to a monster that kills you and it won't be the turn of the hero when this happens in most cases... though it does leave a loophole for New Beginnings if you interpret it literally in a few scenarios. The Hero with the healing Item (not spell) can pass while it's Zargon's or the victim's turn, right? Since the one making the delivery isn't taking an action to do it. This was designed to be an alternative "beginner" quest so it makes sense that the rules here would be a little more lenient. Then again some gaming groups may decide that this is "the rule" until clarified later (it may be a long, long time before they get to EQP/BQP in their sequence especially if they follow the various 'suggested chronologies" floating about online).

Now I think this is the closest to the intentions of the designers when it comes to the NA quests. Some people don't like this. They think it makes the game too easy. Difficulty is relative, as there are many ways that the EU editions were easier than the NA edition while in this area it was much stricter on player character death. There are many ways to make it more difficult. One is to say that you just can't save yourself from death, period, just like the EU editions. Or maybe say you just can't do it with Potions, or make potions "expire" (similar to the EU editions, can't carry treasure potions from one quest to another). I don't have a problem with this, you do what you like in your games. The death of a hero in most cases is more a psychological blow than anything (I know of heroes who despite knowing they can come back to life with a potion nevertheless will use their potion when they are at 1 BP because they don't want Zargon to have the satisfaction of knowing he actually killed them), because the player can just come back with an identical hero with a new name in the next quest and still have success, so long as at least one other hero finishes the mission (or escapes to the stairs alive) and while loot can be lost, on a replay of the quest new loot can be found (Treasure deck is replenished) including any necessary Artifacts which Zargon is supposed to put back in as special treasures early in the adventure. Of course a person who doesn't like these rules can just change them.

Rogar's Hall (suggested by Baker as another alternative beginning quest or before the main sequence, though later retconned in the opinion of other AH people that it should come later in the sequence) free quest throws this all to the wind by having "Mentor" automatically heal any dead heroes with the only limitation being the stock of Healing Potions from the Treasure deck. This blatantly violates the basic rule, but it's a training quest (and one that is actually meant to end in failure most likely) so all bets are off. As written, there is a severe limitation here in that there are only three healing potions (determined by die roll so they might only give you back 1 body point!). Of course the Heroes also will have the same opportunity to escape death using the two available healing spells as with any other quest using the default spell options. This quest seems merely to be meant to be experienced, because whether they succeed or fail, they move on to the regular quests after with little to show for it but the bruises.

The Japanese edition of the game handles things a bit differently. No last minute saves, but the first time you die, you get restored at a Temple (and return in the next adventure) but lose any gold you had. The next time you're dead for good until someone finds the Talisman of Lore. Like the NA's Elixir of Life, it's a bit vague how it is to be used, but one might imagine that it brings any (or all) dead hero(es) back to life on the spot... or it could be used to bring a dead hero back from an earlier adventure even (though no provision is made for more than four heroes at a time) or simply as a behind the scenes thing that happens between quests to retain the name of the old character that otherwise would have perished. This Artifact (Artifacts are called "legendary treasures" in this edition) stays with the Heroes for the remainder of their adventure. Strictly as written, the Talisman allows you to come back to life over and over again, making you (and your party) essentially immortal, with the only downside that each resurrected character loses their gold (gold in this edition is used to "level up" your max body points... though it doesn't say if this progress can be lost by dying). What is the "range" of this thing? Again like the Elixir of Life, it isn't really clarified and so is up to Zargon (or Grimdead, as the evil GM character is known in this case)'s discretion. One who was more difficulty minded might interpret it to mean that the quest ends but you all get to come back again in the next one without having to change characters rather than instantly "respawning" though I think that interpretation is perfectly valid.

In the end, HeroQuest can be whatever you want it to be, as long as everybody at the table agrees! (Zargon is the ultimate authority, by agreeing to play, you agree to abide by his rules)
