This is a fairly common question from people that haven't ridden since the dawn of the STI shifters. The following is a portion of the e-mail I wrote to a friend who had the same question. He's just getting started, so there may be a bit of info. that's too basic. Also assumes you're using Shimano components. If you've got Campy, it's the same principal with a couple of differences. Hope it helps:

Your left hand controls the front derailleur. This means that it changes the chain from the big (flat road) ring to the small (climbing) ring. I don’t remember if yours has a triple chain ring or a double chain ring. If it’s a triple it just means you have an additional set of gears to work with. When you start in your small chain ring and push the entire shifter to the right the chain moves up to the big ring. When you are in the big ring and push the small lever under the brake handle, you release the chain to the smaller ring or rings.

Your right hand controls the rear derailleur. Think of this as the fine tuning shifter. The left one is for large adjustments in chain length; this one is for small ones. You’ll use this one more often. It works the same way. You have 8 or 9 cogs on your cassette. The size of these cogs works in an inverse relation to the front derailleur. The bigger cogs make it easier to pedal. Push the entire shifter to shift down (make it easier) and the lever to drop into the harder gears. The smaller ones are for flatter roads. You’ll find it hardest to turn the pedals when your front derailleur is in the biggest ring and the rear derailleur is in the smallest ring. The small front ring and large rear cog combination is known as the “granny gear”. Everyone has them and everyone uses them, so don’t avoid it based on the name.

Sorry for the length.