so, you've got several things:

your left shifter controls your front derailleur, your right shifter controls your rear derailleur. the derailleur basically pushes the chain on to another cog. in the front, there are two or three, in the back, between five and ten. the rear derailleur also has a spring-loaded pulley section that pushes the chain down in order to achieve tension - because the chain is running over cogs of different lengths, the pulley part needs to take up the slack.

derailleurs are kind of fragile, because they rely on being properly aligned and whatnot. they can break, and they can be replaced. the cable connecting the shifter and the derailleur can break, and be replaced. the shifter can break, and be replaced. the cogs themselves can get worn down - and be replaced. the chain can break, or get worn down, and be replaced.

follow?

if your stuff breaks and you don't feel like putting up with replacing it, you can go to sheldonbrown.com and learn how to convert your multispeed bike into a singlespeed or a fixed gear.