In a vehicle with a manual transmission, the shifter is located in the center console, and is used to select gears. The shifter interfaces directly with the transmission, but most folks would prefer not to see the innards of their tranny, and most transmissions don't enjoy the introduction of dust, coins, and other debris into their delicate insides. Hence the shifter boot.
The typical shifter boot, based on experience with my Mazda Miata, is actually three boots. There is a leather or vinyl boot that is visible. Its primary function is decorative, and to prevent large objects from falling beneath the vehicle.
There is a large rubber boot beneath this one that adds a second layer of protection from dust and debris. On large vehicles such as trucks or buses, this may be the visible boot.
Finally, there is a smaller rubber boot that seals the actual interface point where the shifter moves the rods inside the transmission. This process is helped by the presence of gear oil, which will escape or become filthy if the boots are cracked.
One should check one's shifter boots for cracking every 50,000 miles or so. Also, if shifting becomes difficult, the problem may be resolved by replacing the fluid beneath the boots and the boots themselves. Use a turkey baster to get it out, but preferably not one you use for cooking turkey.