They are the toothed wheels that your chain runs on. One is attached to your rear wheel, the other to the transmittion countershaft. There are also a few sprockets inside you engine for the timing and primary chains, but most of the time people are talking about final drive sprockets, which are the two I mentioned.
Changing sprockets changes your final drive ratio, the same as putting a different rear end in a car. Bigger rear sprocket = less top end. Common reasons for changing include drag racing (Where there is no point in going any faster than you can in a quater mile, and saving a gear change is a plus), Stunting ( ussually uses really big rear sprockets, top end is not required in the parking lot), and touring, where people want to drop the highway RPM.
Hope that helps