To reach the entrance to the Shattered Throne, you must use one of the portals scattered around the Dreaming City to enter the Confluence. Follow the path to the central room and you should see a giant Taken gateway in a large room, with a Techeun standing in front of it. Approach the gateway and follow the prompt it gives you to enter the Shattered Throne. As of Shadowkeep, this is no longer required as you can enter the dungeon directly from the Director.
When you enter Mara Sov's throne world, you will see a large door in front of you. Have your fireteam go through that door, and enter the labyrinth. Continue to advance until you enter a restricted zone. It should be at one end of a broken bridge. On the other side of the chasm, you'll see a group of Taken Acolytes and a Taken Knight miniboss called a Labyrinth Architect. Kill them, but be wary of the Acolytes. They will constantly spawn numerous Acolyte's Eyes until they are defeated. Once you kill all the enemies and the miniboss, a symbol will pop up.
Travel to the location with the corresponding symbol (there are maps online if you have trouble finding each location). There you will find another Taken miniboss, along with a group of Acolytes. Kill them. Once all the enemies are dead, another symbol directing you to somewhere else in the map will appear. Go there. Repeat this process until you have to go back to the beginning. While on the way from destination to destination, various Taken enemies will spawn to block your path, such as Phalanxes, Captains, and hordes of Thrall. There will also be Taken Hobgoblins sniping you from above. It is not required to kill them, but it is highly recommended as it will make this entire section easier.
Once you have gone to every location, you will be directed to go back to the start, where you will face one final group of Acolytes and a Labyrinth Architect in the form of a Taken Captain. Kill them and jump down the hole that will open up. Keep following the path.
After you get far enough, you should trigger a checkpoint. Jump down, kill the enemies below, and at the end of the path, there will be a boss that you need to kill named Asterion, Set Apart. There will be snipers during this portion as well. Kill the enemies and boss and move on to the next part.
You'll find yourself in a long hallway. On the sides are Taken Hobgoblins sniping you, along with Taken Phalanxes and Knights. Carefully take them out one by one, so that they don't trouble you as you make your way across the hall. Be especially wary of the Knights, as their fire attack can deal serious damage. When you reach the end, turn left. You'll enter a wide-open room with catwalks and a few pillars with platforms here and there. Your goal is to make it all the way to the left side. Taken Ogres and obstacles will spawn on the platforms as you progress. Be careful and take out the Ogres as quickly as possible from a distance. Their eye blasts can easily knock you off the catwalks, which will kill you. Jötunn and a kinetic bow is an excellent combo for this encounter, as two Jötunn shots and one shot from the bow will quickly kill the Ogres. Once you make it all the way to the left side of the room, you will find yourself in front of a barrier.
When you reach the barrier, go through it and you'll see that you are slower and you cannot use your jump abilities or sprint. There will also be a never-ending swarm of Shadow Thrall spawning around the room. Go through the maze and avoid the Shadow Thrall or dispatch as much as you can (there is an orb on a pedestal to the right of where you enter. If you jump, then melee, and do this to travel throughout the hall, you will be able to avoid a lot of damage from the Thrall). At the very end, drop down into the next area.
Follow the path. You will eventually be met with a choice: follow the outside of the rock face or go inside. Both paths take you to the same place. Either way, once you make it to the other side, you will find a large group of Taken Phalanxes. Kill them all to get to the next encounter.
Here you arrive at the first boss, Vorgeth, the Boundless Hunger, arguably the hardest part of the entire dungeon. This encounter has a special type of Taken Wizard known as the Keeper of Petitions. In order to do damage to Vorgeth, you will need to kill the Wizards and pick up the orbs they drop. You will receive a buff called "Petitioner's Mark" that stacks three times. There will also be a 45-second timer on the buff. If it reaches zero, the holder of the buff will die and you'll have to kill however many Wizards you've defeated again. Once you collect the fourth orb, the buff will change to "Petitioner's Burden," which also has a 45-second timer. Extinguish the buff at one of the 4 pillars in the center of the room to start the damage phase. You'll have a short amount of time to deal as much damage to Vorgeth as possible. Once the damage phase is over, the Wizards and ads will respawn. Repeat this process as many times as it takes to kill Vorgeth, although it is certainly possible to do it in one phase. A recommended strategy for this encounter is to use a long-range primary weapon like a Scout Rifle or a Bow to kill all the ads in the room from the safe areas near the entrance. Once all of them are dead, bring all four Wizards to very low health without killing them. Then kill all four Wizards in rapid succession and extinguish the buff to start damaging Vorgeth. While immune, Vorgeth will attack with powerful eye blasts that knock you back and deal severe damage. During the damage phases, he attacks with swarms of Axion Darts. Shooting the darts will destroy them and keep you safe from damage. A recommended loadout for this encounter is any kinetic bow to kill ads, Telesto for the Keepers, and a grenade launcher with the Spike Grenades perk for boss damage. If you're running the dungeon with a group, have someone using Tractor Cannon and/or Well of Radiance or Ward of Dawn to help deal extra damage.
Once Vorgeth is dead, go through the door in the center. If this is your first time in the dungeon or you simply haven't done so before, interact with the statue. Next, jump down a series of platforms and ascend a gravitational lift. The path from there is fairly straightforward. If you get confused, just follow the enemies.
At some point, you'll be given a new objective - "Ascend." You have two options: either go up the stairs, go forward and then jump to the ending right away (the harder path) or turn right and then go through a series of jumping puzzles. Stick to the wall and ascend the lifts to get to the last encounter.
Dûl Incaru, the Eternal Return is the final boss of The Shattered Throne. Once you start the encounter, two groups of Taken Psions will spawn. Kill them quickly so they can't duplicate and overwhelm you. Then kill the three Knights. Each one you kill will drop an orb. Pick them up. This is the start of the damage phase and you only have 45 seconds until you have to cleanse yourself in the middle of the room or fail the encounter and have to start over again. Incaru will occasionally summon a crystal that makes any living Knights immune. Destroying it will revive any dead Knights at full health. Once your buff runs out, you will have to break the crystal to revive the Knights and get your buff back. Repeat this process until Incaru is dead. Whisper of the Worm is a very useful weapon to have for this encounter, as well as someone using Well of Radiance or Ward of Dawn.