They all have to do with water /water vapor in the air. Here's how I think of it.
FOG, can be very thick, or not. It usually happens when there is a change in the weather, such as a sudden cool spell, warmup in winter, etc. The wind is pretty calm when there is fog. You probably can't see individual 'drops' of for. Fog does not fall, but rises from the ground, or rolls in from the sea.
MIST is basically rain, just very light rain with small droplets that seem to hang in the air. The water is generally falling gently from the sky.
HAZE, just means there is a lot of humidity in the air. It is not a form of precipitation. You won't get wet from haze. It generally happens during hot weather. The sky will look more white than blue. Far away things are not very clear, as if they're behind a thin curtain.
Hope that helps.