If you're referring to the Chocobo Companion (that's the one you get that's in your Abilities & Actions window) you have to be level 30 and do a quest in I believe it was either Camp Tranquil or one of the other hubs in South Shroud.

For other mounts, most of them are obtained through the subscription or CE bonuses, however you get the Magitek Mount by completing the main story.