Knowing bicycle anatomy helps you know what that do-hicky is that turns the whatchamacallit on your bike. Of course you can always ask your bike shop but it's kind of cool to know yourself.
And if, god forbid, you need to replace it you will at least know what to look for. Also, you can dazzle your friends with your new-found knowledge.
The images shown below of the road bike and the mountain bike, while very different, have the same basic components. There are a few differences like the suspension fork on the mountain bike or the drop handlebars on the road bike.
Although they are configured differently a frame is still a frame and handlebars are still handlebars etc. The components may differ, to accommodate different bicycle styles and riding conditions, but the names and functions are still the same.
Anatomy of a road bike. Click on image for video
The frame is the main component onto which the other components are mounted and consists of...
top tube - connects the seat tube to the head tube
seat tube - The seat tube contains the seatpost of the bike, which connects to the seat post
seat stay - connects the top of the seat tube to the rear dropout
head tube - houses the fork and bearings
down tube - connects the bottom bracket to the head tube
bottom bracket - houses the pedal axle and bearings
dropouts - the slots in the frame and fork that holds the wheel axles
chainstay - a section of the frame that connects the bottom bracket to the rear dropout
Some other key components are...
fork - holds the front wheel and pivots in the headset when steering
chainring - toothed rings that directly connect to the crank
crank - connects the pedals to the chainring
front derailleur - the mechanism that moves the chain from one chainring to another
rear derailleur - the mechanism that moves the chain from one gear to another
headset - houses the bearings that allow the handlebars and fork to turn
cassette - toothed cog that make up the gears ( attached to the rear wheel)