Europa Universalis III divides the world into a number of geographical regions, which are further divided into Provinces. These regions play a role in the requirements for several decisions and missions, among other things. For example, the mission: Convert the people of Asia Minor calls Byzantium to Christianize said region.
Outside of missions and decisions, the only divisions that have a direct effect on gameplay are the continents. Capitals may not be switched between continents unless the existing capital is completely isolated. Mercenaries may only be recruited on one's home continent, and provinces more than a short distance away on another continent are considered "distant overseas" for tax purposes unless there is a land connection.
Note that many of these regions are sub-regions of a larger one, for example, the "Iberian Peninsula" is divided into Léon, Castille, Portugal, Aragon, and Andalucía. Regions may also intersect each other, for example, most of Russia is in Europe but one province is in Asia. A few regions aren't historically or geographically accurate for gameplay reasons. For example, Anatolia is considered part of Europe (probably to allow the Ottomans to shift their capital between the Balkans and Turkey, as happened several times historically).
The hotkey for showing the Region mapmode is 'U'.
List of regions by continent