There are about many different parts in a bicycle. The main ones are, from front to back:
Front Wheelset - 1 tire, 1 tube (presta or schraeder valve), one rim, (usually) 32 nipples, (usually) 32 spokes, 1 hub, one axle and clamp to hold the wheel to the dropouts.
Fork - Connects front wheelset to the rest of the bike. Comes in either suspended or rigid flavors. (usually) 2 dropouts which the hub clamps onto, 2 fork arms, with the exception of Cannondale Lefty forks, one steerer tube.
Headset - Comes in threaded and threadless varieties. Allows the bike to steer freely, and connects the steering assembly (wheelset, fork, handlebars) to the rest of the bike.
Handlebars - What you grip with your hands and steer with. Also what the brake levers and shifter clamp on to. Connected to steerer tube via the stem. Stems either clamp onto the top of the steerer tube in a threadless headset system, or they thread onto the steerer tube in a threaded system. Handlebars come in flat (XC racing mountain bike), riser (downhill racing mountain bike), or drop-down (road bike) varieties.
Frame - The part of the bike that holds everything together.
Crankset - The front part of the drivetrain. Connected to the rear wheel via a chain. Made up of - Pedals, which you put your feet on and come in flat, toe clip, or clipless varieties. Cranks, arms which connect the pedals to the bottom bracket and the chainrings and are removed with a crank puller. Chainrings, round things with teeth to engage the chain with, transmits pedalling force to the drivetrain. The Bottom bracket, which allows the cranks to rotate freely and connects them to the frame.
Seat assembly - The part that you sit on. Made up of - The seat, cushy on the top, with rails that the seatpost clamps on to. The Seatpost, a round tube which clamps on to the seat rails at the top and inserts into the seat tube of the frame. Clamped into place with a small clamp (or sometimes a quick release) at the top of the seat tube.
Brakes - Used to stop the bike. Come in hub brake and rim brake varieties. Hub brakes work by applying pressure at the wheel's hub, while rim brakes apply pressure to both sides of the rim. Disc brakes and coaster brakes are included in the hub brake classification. As there are so many different types of brakes (eg. linear pull brakes, I won't go into explaining how they work.
Drivetrain - Usually the most complicated part of a bike. On a multi-gear bicycle the parts are - The chainrings, which connect the crankset to the chain. The front derailleur, which moves the chain to different chainrings. The chain, which connects the front of the drivetrain to the rear. The cassette, the collective name of the rear sprockets, which transmit pedalling force from the chain to the rear wheelset. The rear derailleur, connected to the frame via the derailleur hanger, which moves the chain among the rear sprockets, and also includes a pulley to maintain chain tension.
Rear Wheelset - Similar to the front wheelset, the rear wheelset differs in that the hub has an extension on one side for the rear cassette to attach to. Therefore, to keep the tire centered in the frame it is necessary to have the spokes on the drivetrain side pull the rim more than the other side. This makes it harder to build and true a rear wheel than a front wheel.