Feathermanes are a tameable family of winged "hybrid" beasts, introduced in Patch 7.2. Members of the family include gryphons, hippogryphs, owlcats, wolfhawks and wind riders (also known as 'wyverns').
As a Hunter, how can I tame Feathermanes?
Feathermanes aren't tameable instantly - you have to earn the privilege! Here's what you need to achieve first:
You must have completed Power Ascended, which requires maxxing out all traits on the Beast Master artifact, Titanstrike, except for Concordance of the Legionfall - you only need to add one point for that. Your artifact should be at rank 52.
You must have completed the Order campaign for Hunters, leading to the achievement A Glorious Campaign, and then the Legionfall quest campaign, leading to the achievement Breaching the Tomb. The Legionfall campaign will not progress very far until you've completed the Order campaign.
What about my Hunter alts? Do I have to repeat this every time?
No! The Tome is Bind on Account. You have to learn the ability separately on each Hunter, but once you've unlocked it on one you may buy additional copies to send to the others.
Are there level restrictions for taming Feathermanes?
You must be 110 to learn the ability.
Are there Hunter specialization restrictions for taming and using Feathermanes?
No. Any spec may tame and use Feathermanes, but to purchase the tome you must have advanced Titanstrike, which is the Beast Mastery artifact, and be in Beast Mastery spec at the time of purchase. This may take a little more effort for players who exclusively play Marksman or Survival. As noted above, this may be achieved through a different Hunter on the same account.
What happens after Legion?
It's highly unlikely that the ability to tame Feathermanes would ever be removed, though the method for learning it may change once Legion ends. We currently have no information on this.