Mounts were introduced with the Winter Update 2010. Due to their popularity, new mounts are implemented every major update. Currently there are 72 different ones. Mounted characters receive a new look and a bonus of 10 speed levels.
Furthermore, another 37 mounts can only be obtained by buying them with Real Life money. 7 of these were limited offers. There's also 1 mount which is the only non-permanent one.
To toggle between mounted and unmounted you may use the hotkey Ctrl+R. You will automatically dismount if you enter a Protection Zone, but since June 3 2014 you will automatically remount when you leave the protection zone, if you were mounted before entering it.
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 activate some Premium Time. However, mounts purchased through the Payment System can be used in-game whenever you want, no matter if you are a free account or a premium account.
For more in-depth information relating to the mounts available to you, see Guide:Mounts.