Deep Fog was introduced in the fourth generation of Pokémon games (DPPt) as an eccentric weather condition, in that it is the only condition that occurs naturally and cannot be created by a Pokémon.
At this point Fog's effects are three-fold. One, in the field it reduces visibility to nearly zero (similar to dark rooms that required the use of Flash in earlier games). The Deep Fog can be lifted through use of the HM move Defog. Second, when battling, Deep Fog greatly reduces accuracy. Third, Weather Ball remains but gets the 150% power boost as with any other Weather Condition (Castform does not change, either).