Ancient Mana is a new currency added in Legion, which is used to complete quests and buy some vanity items and toys. It can drop directly from enemies in Suramar, but there are much faster ways to farm it.
The fastest way to farm them is by looking for containers which contain a lot of them. Twice-Fortified Arcwinegives you between 100 and 150 Ancient Mana per container, while Leypetal Bloossomgives you between 30 and 60 for each. These are scattered around in Suramar, but below are some of the best places to look for them.
1) The market area (coord:s 50,77)
2) The Square (coords: 60,55)
3) Falanar Ruins (coords: 23,37 - 46,85)
4) Vineyards. This place hos lots of Arcwine.
5) Falanar Basement (coords: 20,35)
If you go to any of these five spots you will probably find a few containers. Once you have looted all the ones you can see, use the add-on Cross Realm Assistto change servers. Now you will be on another server, and there is a good chance that they will be spawned on this server again. So repeat the looting process, and keep changing servers until you have enough!
You can get 5 000+ in an hour by doing this, and it's really easy compared to farming mobs. If you are having problems with the mobs destroying your costume, buy some Illusion Bombs to take care of them.