From Bulbapedia, the community-driven Pokémon encyclopedia.
A region (Japanese: 地方 chihō) is an organized area of the Pokémon world.
Typically, a region will have its own Pokémon League, comprised of eight Gym Leaders. Once these eight are defeated, trainers can challenge the regional Elite Four, which may or may not be shared with another, nearby region. Each one also typically has a Pokémon professor, who can teach youngsters about Pokémon and may even give out starter Pokémon to them. However, there is usually a criminal group operating within the region who uses Pokémon to further their goals.
Though there are many similarities between them, there are some major differences between the regions, such as what Pokémon are located there and what legends there are waiting to be uncovered. Visitors from another region quickly realize that the region they are visiting is far from their home, both in the way of the climate, and in the culture of the people there.
In Diamond and Pearl, Prof. Oak explains that Sinnoh, Kanto and Hoenn (and probably Johto and others) are part of an larger country. This concept is similar to the regions of Japan, which the Pokémon World has been modeled after.