A rain shadow is a dry area on the side of a mountain range facing away from the prevailing winds. As moist air is pushed up the windward slope, it expands and cools, and because cool air holds less moisture, clouds form and drop rain or snow on that side. By the time the air crosses the peaks and sinks down the far side, it warms up, its clouds break apart, and little precipitation is left.
The result is a wet, cool windward side and a warm, dry leeward side, and some rain shadows are dry enough to become deserts. Death Valley in California and Nevada sits in the rain shadow of the Sierra Nevada, and the dry Tibetan Plateau lies in the rain shadow of the Himalayas.
Source: National Geographic Society: Rain Shadow (encyclopedic entry)
Image: Rain shadow diagram by Kagee, public domain, via Wikimedia Commons.