Praise the sun. Our Dark Souls Walkthrough provides tips for surviving the adventure, a guide to the entire world, and hints for slaying bosses.
The Demon Ruins are another area that you shouldn't visit until you have obtained the Lordvessel from Anor Lando. If you head into the ruins ahead of that time, know that you won't be able to complete them because of a sealed opening.
When you first enter the ruins by way of the hill that descends to the lowest point of Quelaag's Domain, you'll see a wide ledge ahead of you. Along the ledge, which descends toward lava, there are numerous slow-moving enemies with the eggs on their heads. They're easy to avoid as you start forward and almost immediately come to a bonfire.
Use the bonfire, then start forward through the area. You'll find that the slope that you're descending along descends toward a lake of lava. That lake hides much of the level that you'll need to eventually explore, but you can't pass through it yet because the lava is harmful. As you proceed along the slope toward the lava, watch from some rubble to your right. Just after the rubble, there's a portion of the path that doubles back on itself and you can follow a ledge in that direction to reach a mist-shrouded door.
Step through the mist-shrouded door and you'll find yourself walking along a new ledge that is lined by stone columns. Start forward along that ledge and climb a short ascending staircase. Watch toward the right just beyond the top of that staircase and you'll see a ravine opening leading in that direction. You can head through it to reach the chamber where you'll face the first of several boss encounters in this area.
When you first pass through the ravine that leads into the chamber where Ceaseless Discharge is waiting, there won't be a life meter and you'll only see your foe as a hulking beast in the distance, just ahead of the ledge you are positioned along.
Start right as you reach the edge of that ledge and you'll come to an altar with a corpse draped across it. Check the corpse to obtain the Gold-Hemmed Black Hood, Gold-Hemmed Black Cloak, Gold-Hemmed Black Gloves and Gold-Hemmed Black Skirt. As you loot that corpse, your activity will draw the attention of Ceaseless Discharge.
Once you have the demon's attention, you need to backtrack. Sprint back the way you came, first along the ledge and then (now, as you come from the direction of the altar) left through the ravine. You will appear on the side of the rock formation that is nearest the mist-shrouded doorway that you most recently negotiated. Look to the right as you let your stamina refill and you should see your foe heading around the bend. You want to make sure that he's following, but you don't want to let him get too close.
When you can, run left toward the mist-shrouded door, which now has the mist covering it again because you're locked in combat. When you get there, turn around and watch Ceaseless Discharge, with your shield held ready. He may have decided to turn around if you got too far ahead of him. If that's the case, start forward and he'll turn around again and resume pursuit. Otherwise, wait with your shield ready.
The attacks Ceaseless Discharge will use most resemble those of the hydra monster that you likely battled in Darkroot Basin. He uses some powerful fire attacks, but you should be out of the range of those and should only have to worry about his tentacles. He'll slam those down on the ledge and they'll remain there for a time. You can hack at one (be careful not to get too ambitious and fall from the ledge) and deal damage without much risk from subsequent attacks from the monster.
As you continue to deal damage, you'll either drain his life entirely or Ceaseless Discharge will fall into the lava because he has followed you too far. The latter is the most likely scenario. Either way, the fight should soon be won.
Once Ceaseless Discharge is defeated, the lava lake cools and you can head back through the doorway to the ledge that you had been descending toward the boiling lava. Now you'll find paths that you can follow. Return to the bonfire overhead and rest up if you need to, then start your descent into the cooled lava field.
Descend along the path and ahead of you, there's still a lot of lava left. You should start toward that lava, but for now you shouldn't try to cross it. You can return later for treasure when you have defeated the area boss and claimed an item that will allow you to cross lava. For now, ahead of the first large pool you should turn to the right and make your way toward the wall. There, you'll find the Soul of a Proud Knight on a corpse. Grab it, then look around nearby to find a path that descends toward the right.
As you head along that path, you'll face enemies that look like miniature versions of the Capra Demon you battled in the lower portion of Undead Burg. At first, it's easy to battle only one at a time. They walk back and forth and you can usually lure one toward you, then back away so that you find a more favorable place to fight without luring a second one toward you. That gets more difficult as you continue to progress, though, and these demons hit hard. You want to make sure that you're blocking and countering with the utmost of proficiency, or that you're rolling back and saving stamina (even better) so that you can then counter.
The miniature Capra demons aren't your only enemy in the area, either. After you have taken out a handful of them, you'll find what looks a lot like a floating egg at the top of a staircase. It is capable of using more explosive attacks, but you can still take it out with attacks from the flank. Then descend the stairs it was guarding and head sharply back to the right. As you go, collect two Soul of a Proud Knight along the way and watch for another miniature Capra demon or two.
Ahead of you, there's a campfire positioned near a wall, but a giant centipede head protrudes from that wall. If you try to head directly for the fire, it'll probably cause you a lot of harm. However, you can roam around to the far side (safe from its attacks) and then chuck fire attacks at it. You might suppose that in this hot area, fire attacks would do no real good. That's not the case, though. Your fire attacks will take down the beast easily and then you can use the bonfire that it was guarding to recover energy lost in your recent descent.
Once you're appropriate recovered, you can start back the way you came but you won't want to climb stairs back up to the upper level. Instead, continue forward near the base of the stairs, further along this lower area. Battle any of the floating gas balls you encounter and you'll soon reach an intersection in the path where you can either head left or proceed straight forward. There is a large fellow here with a huge ax who will attack, so try to back away and hit him with magic-based attacks if possible to quickly eliminate him without getting in close. Otherwise, melee attacks will work fine.
On your initial visit to that intersection, it's worth heading left first. Along that route, however, be prepared for a lot of resistance. You'll face another of the huge guys with an ax, and past him the stone pedestal that leads to the left is protected by numerous groups of the caterpillars that will drop from the ceiling to attack. You may find it easiest just to roll past them, though be careful of their toxic spewing that can easily break your weapon. Past a group of four of those caterpillars, you'll find a chest that you can open to obtain the precious Large Flame Ember.
Backtrack to the intersection, where you can look ahead to see a mist-covered door. A final detour that you might consider is the short jaunt uphill (guarded by another of the caterpillars that uses its height advantage to routinely break your equipped weapon if you get even remotely close) to check a corpse near the top. You'll find the Soul of a Brave Warrior on that corpse, if you make the trip. Otherwise, it's time to head forward along the main path.
Ahead of the mist-shrouded door, look to the right. There's a hole in the wall with roots leading down to a lower chamber. You have an important decision to make here. You can descend along those roots on the left side of the room (there's also a Soul of a Brave Warrior on the upper right side) to find a sealed door that only opens if you your Chaos covenant affinity up to a high enough level, or you can keep to the upper path that you were following and pass through the mist-shrouded doorway to face the second boss battle in the Demon Ruins. If you go through the trouble of following the lower route, you will need to battle nine bug enemies as you start through the sealed door. Make sure to kill them all without advancing too far, and then when they are dead it is safe to enter Lost Izaleth. Skip to that section for further assistance with the area. Otherwise, we'll assume that you're planning to work through the Demon Ruins by way of the conventional route that leads through the mist-shrouded door.
Through the mist-shrouded door, you'll find a boss. He may have a new name and new ambitions, but the Demon Firesage is essentially the Asylum Demon all over again. He still relies heavily on his hefty ax, which you'll want to roll out of the way to avoid. He also leaps into the air and floats around, then tries to drop on you and deal crushing damage.
Make the fight easier on yourself by equipping gear that has the highest defense against physical attacks, which is basically all you have to worry about. You can do more damage if you attack with two-handed weapons and eschew the shield once you get in close (it's not really needed here anyway, since your opponent moves slowly enough and gives you time to see most of his moves coming).
Attack from the flanks or belly and hit until you see your foe floating, then roll out of the way as he crashes down on the ground. Move in from there and repeat the process. It's a relatively easy fight, overall.
After you defeat that enemy, you will receive the Demon's Catalyst as your reward. Then you can proceed to the back side of the chamber that it was guarding and continue your journey through the ruins.
You'll come almost immediately to an area with a black opening ahead of and below you, while to the left there are stairs that lead up along the edge of the room. If you head up the stairs, they'll keep climbing and you'll have to battle the gas balls as you make your ascent. They're easy to defeat if you step around to their sides and let them spew their flames in a direction you're not located as you hack at their sides and backs. Note that three of them are guarding a Soul of a Brave Warrior in a recess as you ascend, but those three can be avoided easily if you don't care about their loot. You can keep climbing to reach the top of the area, where stepping onto a lift will take you to Quelaag's Domain near the chamber where you can join the Chaos Covenant.
After activating that extremely handy shortcut, return to the area with the gas balls and descend to the opening you passed previously. Go ahead and descend through that area now and you'll quickly reach a bonfire, which you'll want to activate and probably kindle a time or three. Then head forward again.
As you start through the room, you'll be descending on a massive root. The first thing it will do is curve toward the right side of the passage. From that first bend, you can drop down to a ledge and head along it a short distance to find a corpse. Pillage the corpse to obtain the Soul of a Brave Warrior. Then head down the root to the base of the room. On the far side of the chamber, there's a mist-shrouded door through which you can pass to reach the area's actual boss.
This battle takes place in an arena with a lava bed as a floor. There are cool patches of ground along the sides and those are where you'll want to stay to battle the demon. Otherwise, your footing will work against you as much as your actual foe.
When the fight begins, the demon will be positioned at the center of the area. It may take him a moment to actually start toward you. Be sure to block any attacks that he throws your way from a distance. As he gets closer, he'll hit with some ferocious attacks that can eat up most of your stamina or cause a lot of damage to your life meter if they connect.
Once the Centipede Demon is closer at hand, though, the battle is mostly won. His attacks have decent range, true, but they tend to go over your head if you're nestled in close to his flanks and hacking away at him with your weapon of choice. Let that be your strategy and make sure that you keep an eye on your life meter. You'll probably need to guzzle a few potions throughout the fight. Keep it up and you'll win.
When you defeat the beast, you should receive and equip the Orange Charred Ring. It significantly reduces the damage that you take when crossing the large lava flows in this area and the coming area. You can backtrack to the start of this area now and find the Chaos Flame Ember by descending the first hill and then heading across the lava that first (mostly) drained when you defeated Ceaseless Discharge. At the edge of that area, guarded by several demons, you can find the item. Then return to the room where you battled the Centipede Demon and exit through a narrow shaft along its left side to reach your next destination: Lost Izalith.