If you're in the market for a new bike, the best thing to do after reading this primer is to head to a bike shop. You need to be fit for the frame, and a reputable shop will have the tools and experience to ensure a good fit. Most bike shops will let you take a bike around the corner, and some will even let you take one for a good long ride.
The following descriptions will allow you to read a spec sheet from any manufacturer and understand how each component affects the bike's performance, weight and value. We focus on mountain bikes because of their popularity, durability and comfort, and with a dozen-plus subcategories from each of the major manufacturers, there's no reason you can't find something that suits you.
XC bikes aka trail bikes are set up with lighter-weight components that make long, cross country rides fun. All-mountain bikes are heavier duty but light enough for hill climbs and suspended for descents they're the "do-all" category. Freeride bikes are highly technical rides set up for speed, jumps and tricks. Downhill bikes are demonic machines made for huge downhill drops and speed. Think backcountry heliskiing on bikes cliffs, rock gardens and all.
Make sure you get the frame, forks, wheels and brake system you want at the outset. You can replace just about anything else on the bike if you decide you don't like it later.
A lot of the technology described here also applies to other bike styles, so you're not out in the cold if you're looking for your first road bike.