In games with the Area Control mechanic, players earn points and/or some other advantage for having the most 'control' over areas of the board. 'Control' can be represented by possession of the most resources, placement of the most units in an area, majority of certain cards, and many other themed variants based on the game.
Classic American examples include Risk™ and Axis & Allies™. While many of the games in the library's collection have Area Control as a mechanic, some of the ones where this is a primary mechanic include Power Grid and Acquire.
Due to the nature of the decisions required to gain and maintain control of the 'areas' in these games, critical thinking and problem solving skills are frequently exercised.