Mounts allow you to travel faster than running.
There are two types of mounts:
Flying mounts allow you to fly and are race-specific. You can attack or dig items while being on a flying mount. Flying mounts, also called 'Aerogear', use a special equipment slot. You can attack while using aerogear, although there is a damage modifier. You cannot jump or swim. The Winged Elf Flight Mastery skill only affects their default wings, not any other wings they purchase.
Ground mounts take up a space in your pet bag, and require food and gain loyalty like any pet. They also gain levels and grow slightly faster over time. While on a ground mount, you can single jump and run, but not attack, fly, or swim. Mounts are not limited to any specific race.