Lets start with the cold front. A cold front is where a cold air mass, which is more dense, pushes up a warm air mass, which is less dense. The warm, less dense air gets pushed out of the way by the denser colder air. The weather associated with a cold front is usually, fast-moving, aggressive thunderstorms, followed by a high pressure system and generally cooler temperatures.
The warm front is when you have a warm air mass that overtakes a cold air mass and settles on top of it. The precipitation produced by a warm front is weak, drizzly precipitation, followed by warmer temperatures.
The occluded front is the only front that involves three air masses. Things proceed similar to the cold front above, with a cold air mass meeting and displacing a warm air mass, but then meets an air mass that is in between the cold and warm air temperature. Occluded fronts can produce lots of precipitation, although not as strong or aggressive as the cold front.
The stationary front, as the name suggests, is when a cold air mass meets a warm air mass, but little to no movement occurs. Stationary fronts can sit in the same location for days, producing precipitation similar to that of the warm front.