A shuttle runs between two or more places regularly. The benefit to staying at a hotel close to the airport is that you can take a shuttle, or a regular bus that travels back and forth, to catch your plane in the morning.
Airport shuttles are often buses, although they can also be trains, vans, or any other form of transportation. Some airplanes are shuttles, flying passengers on quick trips. You can also use the word shuttle as a verb, to describe what the bus itself does as it moves back and forth. The original meaning of shuttle is the part of a hand loom that moves to and fro in weaving.