The Dream World is an astral plane best described as the collective unconscious of all Pokémon throughout reality. All mons form an astral projection here when they sleep, but require something to wake them up inside to be fully aware of the Dream World. Humans, on the other hand, are normally unable to enter the Dream World except if they manage to find it while in mon form. After that point, they can enter regardless of physical form, and can even switch between human and mon forms at will. (Though this is highly discouraged.)
The Dream World also connects to the Spirit World, and with the proper knowhow, spirits can be summoned to the Dream World from there, as was the case with Kat's father, who died after getting run over by the Magnet Train.
If a sleeping Pokémon dies in the Dream World, they simply wake up.