There are 25 Mounts that you need to tame in order to unlock this achievement.
Video with all the Mounts:
Anemone 0:00
Sepia 0:27
Bolt 1:13
Everlast 1:54
Indika 2:23
Muse 3:07
Laurion 3:33
Tyrian 4:17
Cypress 4:55
Alabaster 5:35
Itea 6:11
Larkspur 6:42
Guardian 7:17
Sheena 7:48
Quicksilver 8:23
Antikhytera 8:53
Khroma 9:26
Krystalline 10:08
Paragon 10:34
Xenippos 11:11
Krater 11:43
Aethon 12:10
Hellebore 12:35
Asphodel 13:11
Ptilon 13:41