Fog, smog and mist. Three seemingly similar things that are actually quite different - or at least they are for one third of the words. All that most people need to know is that they all impair visibility, and consequently are rather annoying. For a more in-depth definition, however, you'll need to look into the science of each.
Fog is, in essence, a cloud that has decided to settle very close to the ground. Fog is formed when water evaporates from a major water source - lake, ocean, what have you - and condenses in the air to form a soupy haze around the area. Fog typically shows up during hot, wet weather, often coming on the tail end of rain fall where a lot of water has fallen in a short period of time, evaporated and then condensed again to form a cloud long before it reached the sky. Fog's visibility is quite low, and is typically set at a kilometer or less.
Mist is quite similar to fog in that it's activated via the same process of evaporation and condensation, and usually occurs on hot days, often after rain fall. The difference between the two is one of visibility: whereas you can only see a maximum of a kilometer away or less in fog, mist allows for a great deal longer range. The definition is primarily one to help with transit-related visibility, as fog is substantially more dangerous than mist for drivers and pilots.
These two are generally not man-made. They're natural byproducts of the weather. So what's smog?
Smog is a byproduct of humanity. It is the result of the burning of fossil fuels, usually in large plants or in automobiles, which creates airborne contaminants which react with the atmosphere. The result of this mixture is a thick, rather soupy smoke concoction that fills the air with pollution. Smog is most easily noticed on hot days, though it is an ever-present risk in large cities and even smaller communities that utilize internal combustion engines.
At the end of the day, the big difference between these three is the fact that fog and mist will only affect your eyes. Beyond visibility they are harmless. Smog, on the other hand, is a mixture of atmospheric fumes from any number of sources that can harm your body in sufficient amounts. This fact makes smog rather less popular a hazard than the other two, and something environmental groups across the planet fight on a daily basis.