In many (most?) video games, there are failure conditions of some sort. For example, in games where you control a character and there's combat, if the character dies, that's a failure condition. The question here is, how should the game handle it?

On one extreme, there's roguelike-style permadeath, where on death your save file is erased so you have to start all over. On the other, there's games like Celeste where you respawn instantly at the start of the room you were in (or at the last checkpoint in games where checkpoints are really common, like VVVVVV) with no penalty. And, of course, there are plenty of in between approaches.

So, how do you prefer failure states to be handled in videogames?

The quasi-standard way.
As in, you die and the game either reloads the last automatically/manually created savegame itself or by manually reloading a savegame from the main menu.

I don't think there's any one definite answer to this, but I do find that as infuriating as it can be, I do prefer it when death (fail state) in a game feels like it matters. As long as the game is somewhat fair about it, permadeath can make the game exiting and makes you think about survivability a lot more. If you can solve a situation by just throwing lives at it and not really learning anything, if there's nothing on the line, it feels unrewarding.
I also find that a Dark Souls handles death well. You mostly just lose a bit of progress and rarely anything you can't recover (Dark Souls 2's soul memory being the exception really). Death is rarely truly punishing, but you also generally don't want to die. I like it.
But it really depends. Sometimes a game that lets you off easy or doesn't even have a fail state can be good. Perhaps developers should experiment more with the concept.

Yu sort of answered your own question. If VVVVVV had permadeath it would be basically impossible, and if a roguelike respawned you like VVVVVV, it would fundamentally alter the game (and also frequently send players into inescapable death-spirals if e.g. they starve to death, and keep respawning without food)/ There's no one-size-fits-all answer

What about, when your character dies, you die as well.

I accept any failures conditions but I find it annoying when the game has to reboot right after you fail and forces you to go through all the opening screens before you can reload your saved game.

I clearly prefer the "load last save" approach (even automatically, since that's what I'd do anyway). I'm not a fan of perma-death. I get why it's appealing to some though.
I also find respawn mechanics immersion breaking most of the time, unless there is some ingame reason given (like being "the avatar").

For shoot 'em ups, I prefer respawning on the spot with some chance to regain most or all power ups until game over, where there are instead frequent checkpoints. Either that or a rank system (dynamic balancing) where things become easier after death.

Perma-death is ok in SRPGs and where it's possible to keep going if someone dies. I haven't liked it in any rogue-like I've played because they've had too random difficulty curves.

In general saving anywhere is good though not if it affects the difficulty balancing of the game to where there's more trial & error than usual. Same with very frequent checkpoints.

VVVVVV does have permadeath...

on No Death Mode.

(Note that one of the trinkets is easier to get on No Death Mode because the original method of obtaining it requires a death warp, which is not an option on No Death Mode for obvious reasons.) Could handle it like Syobon Action's "Mystery Dungeon Mode" handles it; when you die, you respawn at the last checkpoint, but you get a new randomly-generated level to play through. (Then again, this mode does have a tendency to generate non-clearable levels, so often you just keep dying until the game generates a level that you can clear.) I don't mind trial and error if it doesn't take long to try again.

What I *do* mind, on the other hand, is when you are punished for the "error" part, or when you have to wait a while for each attempt.

There are some issues with this approach:
* Some players still want to get the high challenge and keep trying, and don't want the game to go easy on them. (Hence, this should be optional.)
* There's also the fact that some players may game the system, intentionally playing badly in order to make a later segment easier. You might even see this in speedruns, where a death saves time by making the next part easier in a way that makes it faster.
* Also, this assumes that the game follows the rule that a higher difficulty should not be easier than a lower one; there are games that violate this rule. (For instance, in Touhou 6, to my understanding, Cirno's first spell card is easier on Hard than on Normal. Granted, it's not Icicle Fall Easy (which it literally is on Easy difficulty), but it's still easier than the lower difficulty.)

Well what I mean is where a lot of memorization is required because they didn't think it through enough and just let you save anywhere as a quick fix. So you take a lot of cheap hits and deaths.

For shooters those players could go for a no death run or it could be toggled, but I do like the other system I mentioned better I suppose.

Gaming the system would be obvious from recorded footage so if you're playing to show off your run it would be noticeable.

Yes of course higher difficulty should be higher difficulty. :)

The way you handle death/failure has to be well matched to the type of game it is. It's the mismatch that's most frustrating to players.
Mostly.

I still simply do not like the kind of game like Pac Man where the goal is to stay alive as long as possible facing screen after screen or wave after wave of increasing difficulty or increasing speed or increasing number of monsters or bullets or what have you. It's an arcade mechanic largely based in the desire to maximize quarters into a machine. To my mind, no vestige of this kind of mechanic should endure.

Also, in games where perma-death makes sense - mostly roguelikes - I don't like the ones where you just make it as far as you can and if you die you are entirely back to square one. In the original Rogue what you gained from each run was presumably knowledge - not of the map but of how things worked. You learned things that helped you survive longer on later runs. This makes sense to me. In contemporary roguelikes often there are bonuses or unlocks or traits or something that is actually passed on to your next character. This, too, makes sense to me. Done right, then, perma-death is not actually loss of progress. It only sucks if you died due to a slip of the finger.

If it's using checkpoints, automatically reload the last checkpoint. Similarly, if it's a game where death is unusual (such as point-and-click adventure games), automatically reload the last safe state, as close as possible to where the trouble begins.

If it has manual saving, show the load save game screen as fast as possible. It's frustrating when you are trying to beat a difficult sequence or fight and every time you die you have to watch a slow animation / death screen.

Permadeath

No allowance to load a saved game with the dead character.

Depends on the genre. I can only speak for the genres I that play most often.

* Rouge-like

Perma-death with no carry over is my preferred method I like to see.

* Brawlers

If the game has multiple endings, finite credits towards different endings.
If the game has only one ending, endless credits with enemies getting their health/barrier bar replenished every time you come in.

* Fighting

Not sure if this applies at all here. Some fighting games have a "tour" or "RPG" mode, I would like for it to be more like Fantasy Strike when you get a run with upgrades but if you die you lose it all and start again.

* To Down Shooters

Perma-death with no carry over.