Road Bikes
Road bikes, bikes with skinny tires, are a vanishing breed. For the longest time road bikes, with drop-down handlebars, were essentially the only kinds of bicycles you could acquired. Now road bikes are being eclipsed by the more useful mountain bike. Even so, road bikes still have their nitches. Most racing with bicycles are still done with road bikes. Road bikes still have a slight speed advantage over mountain bikes, although recumbents with full fairings will beat road bikes rather easily. Road bikes are plentiful in second hand markets, so you acquire these bicycles very cheaply.
Mountain Bikes
Road bikes could be considered one end of a continuum. True mountain bikes, also called all-terrain bikes (ATB) are at the other end. The main different is the thickness of the wheel. The thicker a wheel, the better it grips in rough terrain, but suffers from more rolling resistance. Those in between -- bikes with low rolling resistance on pavement, but good control on rougher surfaces are considered hybrid bicycles.
Comfort bicycles are mountain bikes with comfort in mind. When the mountain bike boom started, the mountain bikes were geared to racing, often making them uncomfortable. Comfort bicycles reverses this trend.
http://www.littletoncyclery.com/html/bike_types.html
More info on bicycle types
http://beauty.nagog.smasher.net:81/bike/index.shtml
http://www.jimlangley.net/crank/howtobuyabike1.html
http://isc.faqs.org/faqs/bicycles-faq/
Bicycle FAQ
http://www.transalt.org/features/goodcyclist.html
Bicycle accidents
Mountain Bike Specializations
Stunt Biking
Downhillers | Technical
BMX
Flatland, trail
Freestyle
BMX Crusier
http://www.genesbmx.com/FAQ.html#[-019-]
Tandems
Tandems are bicycles made for two or more. Tandems are a great alternative to two separate bicycles. Two people of unequal ability no longer have the problem of a faster rider waiting for a huffing and puffing slower rider. Essentially a tandem equalizes two partners.
Recumbents
Recumbents are the future of the bicycle market. See the section on recumbents.
Practical Bicycles
City/Commuter, Cruiser, Cyclocross, Electric, Folding, Freestyle, Industrial (For Moving Loads), LowRider, Touring, Track, Trials
Unusual Bikes
Special bicycles are made to handle unusual environments such as water, air and railroad tracks. See the section on Unusual Bikes