The first mounts were introduced with the Winter Update 2010, but new mounts are being implemented at every major update. Currently there are 33 different types. Mounted characters receive a bonus of 10 Speed levels, apart from getting a different look.
Mounts are obtainable by mostly using Taming Items lootable from creatures or rewarded from quests. You might have to use the item on the mountable creature several times to get the mount, and it may break. If you have problems with something visit this FAQ.
Free Account players can tame creatures and earn the achievement Natural Born Cowboy but they will only be able to ride Rented Horses and not any of the "permanent" mounts unless they have active Premium Time on their accounts.
To toggle between mounted and un-mounted you may use the hotkey Ctrl+R.