The Taken War: Earth is one of Destiny: The Taken King's new missions, which you'll undertake once you've completed the main storyline involving Oryx.
The quest chain begins once the main story ends, and involves first completing a mission called "Blighted Coven," which takes you to the Cosmodrome. At one point, you'll be told to kill Taken Champions while on Cosmodrome Patrol. Here's where you'll find the three Taken Champion targets.
As with all Taken events, your targets will spawn every five minutes in an event that says "Taken forces are corrupting the area."
Kill any Taken until it says "A Taken Lieutenant is corrupting the land." Look around the Taken spawns for an enemy with a yellow healthbar. You'll need to kill three of these, which will prompt the message "A Taken Lieutenant is slain."
When that's finally done, you'll see a message that says "Darkness rises! A Taken zealot arrives to claim you." That's your cue to kill the boss.
Irxori, Lost to Oryx – Skywatch
Go towards the building with the radar dish, which is to the Western side of the map from where you first spawn into the Cosmodrome. Enter the Lunar Complex and go all the way through until you emerged at the other end in Skywatch.
Killing him is easy. He doesn't have much health, so just dispatch him from a distance and don't get close or his shotgun will make short work of you.
Stirok, Banner or Oryx – Rocketyard
The hardest of the three. It's possible to do it alone if you focus your fire on him, but you'll probably need two or three people in a Fireteam to kill him and deal with the adds.
To find him, go past The Divide and keep following a roadway to the right. It's basically the next area over.
Stirok packs a lot of health, and summoning him can be just as difficult because of how spread out the Taken Lieutenant spawns are, with each location obscured from the others.
The first Taken Lieutenant area is by the area where you first enter the Rocketyard. The second is past the first tunnel. The third is to the far side. Have each player in your Fireteam camp these locations.
When Stirok spawns, unload into him immediately. You'll need to, if you want to get his health down to zero before he disappears.
If he spawns in the shallow pool of water where the two knights usually are, you might have to deal with them too, or they'll bombard you with AOE damage.
Whatever you do, never stop shooting at him, and always aim for the head.
Darnu, Horn of Oryx – The Divide
Darnu can be found in The Divide. From the spawn area, go into the building with the staircase (Dock 13) and go all the way through and you'll emerged in The Divide.
Darnu's easy, and if you're lucky, you'll even find other players in the area farming for Spinmetal. Once you deal with the Taken spawns, he'll usually appear in the area where the Fallen Devil Walker (spider tank) event usually happens.
Avoid his black bubbles, which blind you, and kill him.
