I will answer as objectively as possible:
Fainting (FNT) (Japanese: ひんし near death) is a status condition in which a Pokémon is no longer able to battle.
In the Pokémon anime, fainting is referred to as being unable to battle (Japanese: 戦闘不能 unable to fight). These Pokémon are not always unconscious, but merely do not have the energy, ability, or other reasons to continue battling.
A Pokemon you faint has not died, rather (as fainting implies) the Pokemon is knocked out. In the Anime, when Pokemon are unable to battle, they are often out of energy and can no longer fight for that reason.
Furthermore, Japanese refers to fainting as "near death", meaning not dead, but harmed in some way. In some other languages, fainting is called "fake death", meaning the state that they are in isn't death, but deathlike- their heart rate is low and they are unconscious.
In theory, if a Pokemon kept harming them whilst they fainted, they could die, but that is too cruel for any of the ~10 year-olds we play as in game to ever do (unless you are a psychotic sociopath; luckily, the games, being kid's games, do not allow you to do such a vile act of heresy).
So, when you faint a Pokemon, they are most likely out of energy or unconscious. Luckily, those Pokecenter Ladies are expert nurses with super advanced medicine and technology to heal all of your Pokemon's wounds, broken bones, and brain injuries (or at least, I hope that's what they do...). For wild Pokemon, well, they have to survive the wounds you inflict them to survive. If they can get over the wounds, they survive, if they can't, they die (basic Darwinism).
Hope I Helped! :P