Which side of an island is wet is directly related to the normal direction the wind seems to come from. This 'normal' wind direction is known as the trade winds in the tropics.
[Trade Winds: Any of a consistent system of prevailing winds occupying most of the tropics, constituting the major component of the general circulation of the atmosphere, and blowing northeasterly in the Northern Hemisphere and southeasterly in the Southern Hemisphere. Often used in the plural.]
As the air hits the island and/or its higher elevations and mountains, it must rise. Rising air expands (air pressure drops with height) and cools. When it cools, then the water vapor in the atmosphere condenses into clouds (1) and then can rain (2). This is known as the windward side of the island.
The other side of the island sees the opposite happening (warming sinking air and no clouds) and stays dry. This is the leeward side of the island.
( these pictures are from the USA Today web site : http://www.usatoday.com/weather/tg/wupslope/wupslope.htm )
The trade winds can weaken or change direction during large scale atmospheric circulation changes like El Niņo.