Chainsaw more or less detailed the styling elements that go into the various types. I'll add this:
A board track bike is patterned after early race bikes that were raced on board tracks. Not bicycles, mind you (early race bicycles looked much like they do today), but racing motorcycles. By rights, these bikes should have some form of tank and a sprung saddle. Often, white or creme balloon tires are used to emulate the look of early natural rubber tires.
A flat tracker is also pattern after racing motorcycles, but from those used on later dirt track speedways. I think this is a less common style among bicycles.
A chopper is patterned after customized motorcycles that became the rage in the '60s and '70s. Raked forks, small front wheels, banana seats, & unique custom modifications typify these bikes.
Muscle bikes are essentially factory-built choppers, typically based on 20" size. Also called wheelie bikes, the combination of a larger rider (then you'd traditionally have put on a 20" wheeled bike) and a banana seat moved the center of gravity pretty far back, so made for easy wheelies.
Bobbers are yet another modified motorcycle-based style. This was something of hot rodding for bikes, people "bobbed" a motorcycle when they stripped off extra weight and cut the fenders short (bobbing them). But typically, OEM tire sizes are kept and there's an emphasis on performance.
Road bikes are typically those used for road racing & training. These are different triathalon bikes, and different still from fixed gear track bikes, and different yet again from urban "city" bikes.
Touring Bikes are road bikes with relaxed geometry, so they're less twitchy and more comfortable to ride for long distances, like cross-country. Typically still have drop bars, these usually are outfitted with fenders, carriers and panniers, but that can vary depending on how much the rider plans to rely on their credit card...
City bikes are typically more upright and utilitarian. The emphasis is one functionality and utility. These are typically used for commuting to work and running errands in a location where things are close enough together that you can get away with it.
Fixies are fixed-gear bikes, with no freewheel ability, and often have no brakes. These are decended from track bikes and common amongst bike messengers and hipsters that think riding an utterly impractical bike makes you cool.
Hybrids and comfort bikes offer different levels of combinations of the features of road bikes, urban bikes, mountain bikes, etc. All are some level of compromise, your results may vary. These work quite well for the majority of casual riders that never conside buying more than one bike...
Cyclocross bikes are a different breed of hybrid. More of a road bike with narrow, knobby tires, optimized for competing in cyclocross competition. These bikes also tend to lend themselves well to all around use. If a road bike is to a sports car, and a mountain bike an SUV, then 'cross bikes are the equivalent of rally cars.
Cargo bikes have modified frame designs, to allow hauling large loads. These come in 2 and 3 wheel versions, front platform, long tail, and practically any other variation you can think of, but are much more often cutom taylored to the application.
And then rat rod bikes. Well, we must all know exactly what they are because we don't have to discuss it!