These Pokémon have a natural toxic quality, maybe they produce venom, are poisonous plants, or even represent pollution. They normally live in caves, marshes or similar places.
A very curious fact about Poison-type Pokémon is that some of them resemble ninjas. For example, Koffing is like a smoke ball, Crobat has a ninja's speed and secrecy and Toxicroak has paws that make it resemble a ninja.
The Poison type is related to the status effects Poisoned and Badly Poisoned, which are produced mostly by Poison-type moves.
They're strong against the Fairy type because fairies represent purity and poison the pollution.
They're strong against the Grass type because trees and plants are damaged by the pollution and toxins, such as insecticides.
They're weak against the Ground type because earthen and sandy features neutralize and mitigate the toxicity of poisons, making them less effective.
They're weak against the Psychic type because mind can't be poisoned. Poison acts over the body, and the mind is superior to the body, while Psychic-type attacks are mental.
They resist the Fighting type because when poisoned, a fighter or deportist loses strength. Fighters try to be healthy as most as they can and poisons can make them sick. It could also be said that since some Poison-type Pokémon have a gelatinous body, punches and kicks cushions the impact.
They were weak against the Bug type in Generation I because of insecticides, toxins that killed bugs. However, since Generation II, the Bug type resists them because many insects have developed resistance to some insecticides.
The Ghost type resists them because you can't poison someone who's already dead, besides, the spirit doesn't have a physical form, thus toxins can't pass through.
The Rock type resists them because since rocks aren't live creatures, poison can hardly damage them.
The Steel type is immune to them because poisons cannot circulate through or penetrate metals, mainly because they're not organic materials.
There are no Poison-type Legendary Pokémon, excluding Arceus holding a Toxic Plate.
Most dual Poison-type Pokémon have Bug or Grass as its other type. This reflects how in real life many insects and plants are poisonous.