Initial
There are flowers with trapped Sylvari in them throughout the dungeon. More often than not, if you open one you'll get a powerful boon for 2 minutes that will heal you once right before you die. Sometimes, though, you'll have to fight a member of the Nightmare Court. They're easy alone, but make sure you don't open any flowers in the middle of a battle. Try to save them for right before a boss or other hard fight.

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Sariel
Sariel has only one dangerous attack: her knockdown combo. She knocks you down so you can't escape, then finishes the combo with enough damage to down most classes. Dodge the knockdown, have someone else crowd control her off you when you get knocked down, or bring a skill that lets you escape knockdown. However, she's not dangerous outside of that combo, so if you get downed just have someone get you up again.

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Brangoire
Brangoire is, by a large margin, the hardest fight in the dungeon. He doesn't do much damage by himself, but he summons an endless stream of Living Nightmares. These translucent spiders will chase after you. If they get close enough to a player, they will pause for a split second and then explode, creating a cloud that poisons anyone who walks through it. The poison will do thousands of damage if allowed to run its course. Worse, Brangoire will fear characters, preventing them from dodging and often send them running into poison clouds. A skill that breaks fear or grants invulnerability, like Elixir S, is very useful, as are skills that cure conditions. Run away from the spiders and shoot Brangoire while doing so - ranged weapons are a must for this fight. The spiders have relatively low health and don't explode if they're killed, but killing them effectively and dodging at the same time is difficult. Large AoE attacks, attacks that bounce between multiple targets, and area snares like Glue Shot can clear or control a group of spiders so some of the party can focus on Brangoire. Focus on staying alive and you'll be able to take him down. Thankfully, Brangoire has relatively low health for a boss.
Running, circling him and only attacking him and not the Living Nightmares, works really well.
An engineer with Bomb Kit is extremely effective here. Just run a circuit around the room aggroing as many spiders as possible, while spamming the kit's basic attack. If party members have any stealth or teleport abilities, they can drag their spiders over and drop aggro onto the engineer as well.
A mesmer with Time Warp can also help low-damage parties deal with Faolain when she appears to resurrect an illusion. Use it when you are all together to quickly burn her down to 75%.

Watch out for the pack of Nightmare Hounds after Brangoire. They do high damage and are enough to wipe an unprepared team. They're melee only, so keep them immobilized, crippled, or otherwise distracted. Don't be too proud to run if you lose control of the battle.

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Cadeyrn
Cadeyrn is a chance to relax after the string of hard fights earlier. Just pound on him until he dies. If you get turned into a kitten, kill one of the Nightmare Puppies to return to normal.

In the next area, make sure that you only open flower at a time. Don't put down the essence if you pick it up (it will disapear). Most importantly, there's a pack of spiders down a corridor across from the boss. Try not to aggro them while you're fighting something else, and consider walking down the corridor so you can kill them first. You need 4 essence thrown at him at once to shrink him. Once he's normal sized you can fight him.

Bug Bug.Moa Transformation is bugged on this boss. The boss reverts to its initial, invulnerable state when the moa form expires.
 

Faolain
There are four nightmare companions Logan, Eir, Rytlock and Zojja. The party needs to focus one at a time. Once a nightmare champion falls, Faolin will attempt to resurrect the fallen champion. Simply focus Faolian and deal enough damage to stop her from completing the resurrection. If you fail to stop Faolian she will bring back the fallen champion.

The key to success is stay moving. Once Eir is down the threat of being picked off at range is gone. Don't use any skills that lock you in place, timing skills like hammers knock back is the difference between completion of the dungeon and a wipe. Being range or melee doesn't matter for this fight, just be aware what enemy is targeting who.

The only heavy area of effect attacks are from Eir Spread Shot and Zojja's slow moving Lightning Orb.

Use the tree in the middle to your advantage against Zojja's Lightning Orbs.

Lastly, focus a single target and burn everything as soon as you see Faolian. Stop her once and she will not resurrect a fallen Nightmare Champion. It would be better to wipe and defeat one champion than to save your strongest skills for the next one. However, if it's possible to drop another champion while Faolian is reviving an earlier one, she will switch to the new one and leave the previous to die.

Once all nightmare champions fall, the fight is complete.