Area movement means that the game board is divided into areas *of varying size* which can be moved out of or into in any direction as long as the areas are adjacent or connected. A classic example being Axis & Allies, in which land and water are divided into variable areas and boundaries that define connectivity between various areas.
Area movement is one way to handle movement on a game board. Two other commonly used ways are Grid Movement and Point to Point Movement. However, Area Movement is arguably just a form of Point to Point Movement, in which areas act as "points" with implicit "lines" connecting each area to all adjacent areas, as in Risk, which directly uses point-to-point movement for overwater movement.