Celestial Tournament is a weekly quest on Timeless Island added in late MOP. Defeated the pet masters will get Celestial Coin, you can buy four Celestial Pets (Calfling of Niuzao, Calfling of Niuzao  Xu-Fu, Cub of Xuen Chi-Chi, Hatchling of Chi-Ji) from NPC with the Celestial Coins.
Celestial Tournament can be completed once per week (weekly quest) and only by one char on your account (account-wide quest). You have to have 15+ battle-pets on Lvl 25 and char with level 90 for starting the quest. You have unlimited time and number of attempts, but you have to complete it without a break (you lose any progress if you leave tournament scenario). 
"Revive Battle Pets" or "Battle Pet Bandage" are invalid in Celestial Tournament, only pet's healing available. Once your pet dies in battle, means failure, you have to leave the quest and start again. So you'd better do enough preparation before the quest because any mistakes will lead to failure. You have to defeat a group of three pet masters during the first round and four legendary pets during the last round. There are three different groups of pet masters which rotate every week, but legendary pets in the final round (Celestials) are still the same. Be prepared for really hard battles.
You will use all type of pets, like Pandaren Water Spirit and Chrominius, the VIP combination which is the main force to defeat four legendary pets. Perhaps Blizzard thinks that the VIP combination makes a lot of pet battles no tricks, Pandaren Water Spirit will have been significantly weakened in Legion. Maybe the Celestial Tournament will become more challenging that time.
If you don't know how to choose the pets, you can reference the Battle Pets Extra Value Package on our website. The Pets in this Package can defeat all of the Pet Tamers, including the Celestial tournament quest.