Barding is a set of armor or an outfit your chocobo companion can wear. They do not add stats or anything to your chocobo so they are purely for aesthetics. Most sets consist of a head piece, a body piece, and footwear. Below are bardings currently available and the methods to obtain them. Click on the thumbnails to see the full set on a chocobo.

Barding of Light:
Veteran Rewards program, pay for 360 days of game time

Behemoth Barding:
FFXIV:ARR Collector's Edition
(You can pay money to upgrade a regular account to collector's at the Mog Station)

Black Mage Barding:
Veteran Rewards program, pay for 180 days of game time

Dragoon Barding:
Learn all skills in chocobo companion's DPS skill tree

Eerie Barding:
Halloween holiday event
(No longer available)

Egg Barding:
Hatching-tide holiday event
(No longer available, however it is tradeable and can be bought off the market board)

Gambler Barding:
Redeemed for 20,000 MGP from the Prize Claim Vendor at the Gold Saucer (X:5 Y:6)

Gridanian Barding:
6,000 Grand Company seals, must be with Order of the Twin Adder and at least Serpent Sergeant Second Class rank

Gridanian Crested Barding:
8,000 Grand Company seals, must be with Order of the Twin Adder and at least Chief Serpent Sergeant rank

Gridanian Half Barding:
4,000 Grand Company seals, must be with Order of the Twin Adder and at least Serpent Sergeant Third Class rank

Gridanian Saddle:
Comes with your chocobo companion if you're with the Order of the Twin Adder

Ice Barding:
4-star leatherworker craft, requires Diamond Tear from Shiva Extreme trial, an Allagan Leather, a Cashmere Cloth, and a Platinum Nugget (barding is tradeable and can be bought off the market board)

Levin Barding:
3-star leatherworker craft, requires Levin Orb from Ramuh Hard trial, a Saurian Leather, a Rose Gold Nugget, and 4 Twinthread (barding is tradeable and can be bought off the market board)

Lominsan Barding:
6,000 Grand Company seals, must be with Maelstrom and at least Storm Sergeant Second class rank

Lominsan Crested Barding:
8,000 Grand Company seals, must be with Maelstrom and at least Chief Storm Sergeant rank

Lominsan Half Barding:
4,000 Grand Company seals, must be with Maelstrom and at least Storm Sergeant Third Class rank

Lominsan Saddle:
Comes with your chocobo companion if you're with the Maelstrom

Paladin Barding:
Learn all skills in chocobo companion's tank skill tree

Paramour Barding:
Valentione's Day holiday event
(No longer available, however it is tradeable and can be bought off the market board)

Plumed Barding:
Train someone else's stabled chocobo 30 times achievement reward

Sleipnir Barding:
Trade with the vendor at Revenant's Toll in Mor Dhona (X:22 Y:6) for 5 Odin Mantles from the Odin FATE

Sovereign Barding:
300 player commendations achievement reward

Starlight Barding:
Starlight Celebration holiday event
(No longer available, however it is tradeable and can be bought off the market board)

Tidal Barding:
3-star armorer craft, requires Leviathan's Barb from Leviathan Hard trial, a Gold Ingot, a Darksteel Plate, and a Hippogryph Leather (barding is tradeable and can be bought off the market board)

Ul'dahn Barding:
6,000 Grand Company seals, must be with Immortal Flames and at least Flame Sergeant Second Class rank

Ul'dahn Crested Barding: 8,000 Grand Company seals, must be with Immortal Flames and at least Chief Flame Sergeant rank

Ul'dahn Half Barding: 4,000 Grand Company seals, must be with Immortal Flames and at least Flame Sergeant Third Class rank

Ul'dahn Saddle:
Comes with your chocobo companion if you're with the Immortal Flames

White Mage Barding:
Learn all skills in chocobo companion's healer skill tree