Heritage of the Zandalari

When you reach 110 you need to do the introduction quest to go to Zandalari ( or skip it if you did it with a main) and speak with Talanji (she is up on King Rastakan's throne).She will give you a quest and then you get the achievment and the armor

Heritage armor:
The ones of allied races require to level up one character of that race without a level boost (heirlooms are allowed) and gain the achievement (like Heritage of the Lightforged). Once earned, you can use this transmog on every alt of that race.
Older/main races may complete a quest chain to get their heritage armor (level 50+ and possibly additional requirements, depending on their race).
With 10.1, heritage armor quests for non-Allied races no longer requires exalted reputation with their home factions.
In case it doesn't pop up on log in, try to visit the Embassy to find the quest giver (Stormwind / Orgrimmar). A relog might also be helpful if you've just leveled up.
Some races need a teleporter near the Embassy to turn in their quests. /way Stormwind 54.4 15.4 Embassy SW /way Orgrimmar 40 76 Embassy OG

Alliance:

1. 
   
2. Dwarf: [transmog-set=2897] - Questline starts in Stormwind: Keep Yer Feet On The Ground
3. 
   
4. Dark Iron Dwarf: [transmog-set=2765] - The Anvil-Thane's Designs
5. 
   
6. Gnome: [transmog-set=2995] - Shifting Gears
7. 
   
8. Human 1: [outfit=460885] - An Urgent Matter
9. 
   
10. Human 2: [outfit=460884] (you'll get all three colors of the Human's armors. There are also [item=203176]s as alternatives for the chests, like [outfit=486147])
11. 
    
12. Human 3: [outfit=460883]
13. 
    
14. Kul Tiran: [transmog-set=2950] - Heritage of the Kul Tiran
15. 
    
16. Lightforged Draenei: [transmog-set=2780] - Heritage of the Lightforged (use Stormwind teleporter at Embassy)
17. 
    
18. Mechagnome: [transmog-set=11976] - Heritage of the Mechagnome (use Stormwind teleporter at Embassy)
19. 
    
20. Night Elf: [outfit=486135] - The Clarion Call
21. 
    
22. Void Elf: [transmog-set=2782] - Heritage of the Void
23. 
    
24. Worgen: [outfit=233232] - The Shadow of Gilneas
25. 
    

Horde:

1. 
   
2. Blood Elf: [transmog-set=2896] - Questline starts in Orgrimmar: The Pride of the Sin'dorei
3. 
   
4. Goblin: [outfit=228099] - Old Friends, New Opportunities
5. 
   
6. Highmountain Tauren: [transmog-set=2779] - Heritage of Highmountain
7. 
   
8. Mag'har Orc 1: [transmog-set=2778] - Heritage of the Mag'har
9. 
   
10. Mag'har Orc 2: [transmog-set=2777] (you'll get all three colors of the Mag'har Orc's armors)
11. 
    
12. Mag'har Orc 3: [transmog-set=2766]
13. 
    
14. Nightborne: [transmog-set=2781] - Heritage of the Nightborne (use Orgrimmar teleporter at Embassy)
15. 
    
16. Orc 1: [outfit=486143] - A Summon to Orgrimmar
17. 
    
18. Orc 2: [outfit=486144] (you'll get all three colors of the Orc's armors)
19. 
    
20. Orc 3: [outfit=486145]
21. 
    
22. Tauren: [transmog-set=2952] (the big totem is a cloak: [item=170063]) - When Spirits Whisper
23. 
    
24. Undead: [outfit=486131] (you can chose between two chest styles, [item=210054] (3D_1 and [item=208476] 3D_2) - Unliving Summons
    - your Undead must have completed the Dragonflight Pre-Patch event questline Return to Lordaeron. If you still need to complete it, meet Calia Menethil for Call to Lordaeron
25. 
    
26. Vulpera: [transmog-set=11977] - Heritage of the Vulpera
27. 
    
28. Zandalari Troll: [transmog-set=2951] - Heritage of the Zandalari
29. 
    

You can change the previewed race by clicking on the arrow below the picture on the armor's page.

Once earned, the heritage armor can be used by any Zandalari alts, including low level, boosted and race changed ones.