Föhn wind:
Background knowledge - "dry air" changes temperature by 3 deg C per 1000', while "saturated air" changes by 1.8 deg C per 1000'. Dry means anything that isn't saturated, so it can still hold lots of water vapour and be considered as dry. Also the temperature changes or Lapse Rates are sometimes given different values in different text books. The main point here is that dry air changes temperature more quickly than saturated air.
1) When air cools, it can reach saturation. As a result, clouds and precipitation is formed. At this point it cools more slowly. (The reason for this is that energy is released when water vapour condenses to create liquid water.)
2) When air moves towards a mountain, it is pushed upwards (orographic lift). When air rises, it cools. So the process in part 1) can occur.
3) When the air reaches the other side of the mountain, it has given up some of its water, so it is drier air.
4) As this drier air descends, it warms up more quickly than it cooled going up the first side, because it is not saturated any more.
So this air moving down the lee side of the mountain is warmer and dryer. It's called a Föhn wind because it is common in the Föhn region.
Note this doesn't always happen. It requires air of the right temperature and humidity to be moved across a mountain range in such a way that orographic lift occurs. The bit that makes the lee side warmer is precipitation on the upper windward side of the mountain.