Smog is what happens when the words 'smoke' and 'fog' flow together into one killer low-lying cloud. London is famous for its fog, and was one of the first to become famous for its sooty coal smoke. Unsurprisingly, the word was most likely coined to describe conditions in London in 19051, and one of the first big smog-related disasters also occurred in London some 48 years later. (Not the first, though. Read about the Donora Smog of 1948, for an earlier example.)
Smog, these days (at least in developed nations) is less due to burning coal as it is to industrial exhaust and burning gasoline, usually in personal automobiles. Cities have lots of people who drive lots of cars (often not even moving--just sitting in traffic burning up gasoline, unless they have one of those new-fangled contraptions that doesn't) that spew out noxious, smog-causing fumes. (Industrial exhaust also brings us new fun, as weird chemical emissions can be altered by sunlight and then combine with others to form photochemical smog.) Smog is usually a localized problem (urban areas create their own smog--rural areas usually don't get dense enough emissions to cause this particular problem), but national and international regulations are still put in place sometimes. Even if suburbanites don't get a smog cloud, the same things that cause smog in the city also cause problems elsewhere.
To combat smog, many nations now have emissions regulations, requiring relatively clean-burning machines, mandating taller smokestacks (sending the pollutants high into the atmosphere where they can form acid rain that falls someplace else), prohibiting use of optional dirty machines (lawnmowers, for example) during times at high-risk of smog, catchment of emissions, or other various measures. Some cities, such as Mexico City and Stockholm, have opted to tax or restrict car usage in at least parts of the city0, and most have some sort of public transit. (Usually fairly paltry public transit here in the US.)
Luckily, just emitting fumes, which we do all the time, isn't enough to make killer smog clouds, nor will it always produce the much more common bad-for-you-but-no-one-died-this-time smog clouds. Wind will help blow away the pollutants before than can get dense enough to cause a big problem (though they will cause a problem, but over a diffuse, less populated area, one hopes). The warm emissions (smoke is created how?) tend to rise up into the sky where 1. we aren't breathing it and 2. there's more wind to blow it far, far away. This rising action is important. If warm air from somewhere else blows in, it can create a temperature inversion, holding our emissions down near the surface. This is especially easy to happen when your city is built in a big hole. Luckily we don't build cities in holes. Well, except for Mexico City, Los Angeles, Beijing, Santiago... (Okay some of those are just partially surrounded by mountains, but the end result is the same. We get smog.) Build your city on a hill and you'll have less of a problem with smog. (Stop spewing massive quantities of pollutants into the air and things start looking even brighter!)
The cities most famous for their smog are probably London, Mexico City, Los Angeles, and (thanks partly because of the publicity of the 2008 Olympics) Beijing. They're certainly not alone, though. Tehran has had major problems with it. The Asian brown cloud is big and persistent enough to have its own name (and a writeup on E2!)
Sometimes smog-stopping measures aren't enough, so we hide or leave. Some places will prohibit school children from going outside when the smog is bad, and sometimes we go so far as to bus them out of the city for some fresh air time. (Couldn't we just drive less?)
You've probably figured it out, but I suppose I must mention explicitly that smog is bad for people, especially the very young and the very old. It irritates the lungs, the throat, and the eyes. It makes people cough, wheeze, choke, and die.
I live in a city and, while we don't have much smog, there have been times in the morning where I can see and smell the haze. Buses can help. Bikes can help. Light rail can help. Convincing people to use these things, however, is every bit as difficult as building them. Cars are convenient--even if you're stuck in traffic, at least you have an air conditioner/heater, a radio (and maybe a CD player), and your own personal space free of belligerent and unshowered homeless folk. Plus, the smog is out there, while you're in your own airtight bubble. Surely you'll agree that we could solve a lot of problems if everyone else would use public transit and you were the only one left driving. But, of course, those selfish bastards refused to coöperate for your greater good.
0 I initially claimed that London had something like this, but spiregrain points out that my memory is faulty--I can find no such reference. Found some for Mexico City, though: http://en.wikipedia.org/wiki/Hoy_No_Circula and Stockholm (http://en.wikipedia.org/wiki/Stockholm_congestion_tax) -- other potentially interesting reads: http://www.rff.org/Publications/WPC/Pages/08_15_08_Driving%20Restrictions%20and%20Air%20Quality%20in%20Mexico%20City.aspx http://en.wikipedia.org/wiki/Car_Free_Days http://en.wikipedia.org/wiki/Road_space_rationing
Updated 2022-01-24: It appears that London did create the London low emission zone in 2008, and has since added an "Ultra Low Emission Zone".
1 Wikipedia seems to agree with this etymology, but then goes on to claim that the term was used in the Los Angeles Times on Jan. 19, 1893. Citation needed!
For Wintergreen: An Earth Quest.