A driver is a component of a loudspeaker that converts electrical audio signals into sound waves. A driver is really only the transducer part of a loudspeaker but is often itself is referred to as a speaker.
There are many different kinds of drivers but they all do basically the same thing: create sound waves. By far the most common type of driver is the moving coil electrodynamic piston driver, which creates sound with a diaphragm that acts like a piston to pump air. Woofers tend to use paper cones as a diaphragm, whereas tweeters often use a fabric dome.
Drivers come many different sizes because it's virtually impossible to make one piston driver that can reproduce sound waves over the entire 20 Hz to 20 kHz frequency range of human hearing. To produce low frequencies a driver needs to have a large diaphragm and enough mass to resonate at a low frequency; to produce high frequencies a driver needs to have a small diaphragm with a low mass. Obviously, these requirements are in opposition so drivers are usually designed to produce only a portion of the sound. This gives rise to multi-way speaker systems which typically use a tweeter for the high frequencies and a woofer for the low frequencies.
Besides their different size, there is another very significant difference between a tweeter and a woofer. The tweeter usually has a sealed back while the woofer usually has an open back. With a sealed back, the tweeter emits sound waves from the front side only. A woofer emits sound waves from both its front and back side.
Tweeters are the smaller drivers since they produce the highest frequencies with the shortest wavelengths. Woofers are the largest drivers since they produce the lowest frequencies with the longest wavelengths. Other driver sizes include mid-range drivers and subwoofers. Mid-range drivers reproduce middle frequencies and are used in multi-way speakers with three or more driver sizes. Some have open backs and some have sealed backs.