The first mounts were introduced with the Winter Update 2010, but new mounts are implemented every major update. Currently there are 50 different ones. Mounted characters receive a bonus of 10 speed levels, apart from getting a different look.
32 mounts are obtainable by using Taming Items lootable from creatures on them or by earning them through a quest. You might have to use the item on the mountable creature several times to get the mount, and it may break. If you have any questions about taming, visit this FAQ.
Furthermore, 16 mounts can only be obtained by buying them with real-life money. 4 of these were limited offers. There's also 1 mount which is the only non-permanent one, and 1 which is not obtainable as of yet.
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.