Praise the sun. Our Dark Souls Walkthrough provides tips for surviving the adventure, a guide to the entire world, and hints for slaying bosses.
When you enter Quelaag's Domain, you'll find that it's a long tunnel descending into the heart of the webbed hill that you just climbed along the edge of the lower Blighttown swamp. There are webbed cocoons overhead.
As you descend along the shaft, you will eventually come to some strange objects in the corridor. These are actually enemies. You can pick at them from a distance with arrows or fire attacks and you'll break apart what looks like an egg. That turns out to be an assortment of bloody worms. They move quickly and several of them will come toward you now that you've broken them apart. They're actually able of causing some serious harm, so you may find it better to simply run past them. If you do deal with them, fire attacks continue to work well.
Past that threat, the trail continues its descent and stops finally ahead of a mist-shrouded opening. When you step through that opening, you'll reach the area boss.
Chaos Witch Quelaag
Quelaag is a fearsome foe, the most difficult boss that you've yet encountered. She is half giant armored spider, quarter woman. However, her fighting tactics resemble those that you might expect to encounter from a dragon.
When the battle starts, let Quelaag skitter toward you. She will rely heavily on fire attacks, so make sure that you've equipped fire-resistant gear if you have any. If you are standing in front of the spider, it will likely try to hit you with flame attacks. It's better if you can move in close before that happens and stand to either side of its mouth, between the first and second legs on either side of the mouth. From there, you can hack away at the spider and do some pretty good damage, maybe even use two-handed attacks.
Once you have done enough damage from there, the spider will shift away. Keep in mind that if you remain at too great a distance, Quelaag will try to use a luminescent blade to swipe at you. Though you can block part of the damage with your shield, you're still likely to take damage and suffer from stamina loss. You're better off if you for the most part stay close to your foe throughout the fight. Then you only have to worry about the occasional noxious fumes that the spider will send out (watch for her lovely upper body to drape forward if you want to know when the noxious explosion is about to happen), or a lava flow that it sends out to either side of its body. Except when you're rolling to escape a lava flow or the fumes, you're always safer right up against the critter than you are trying to stay at a distance.
Besides having a few powerful attacks, Quelaag also has a larger life meter than other bosses up to this point. You'll have to hack away at her for quite some time to do any real damage. It's a battle of endurance, but staying close should be enough to give you the upper hand and it should allow you to eke out a victory.
When you defeat the boss, you will receive the Soul of Queelag item. You can head up the stairs beyond where the battle took place and when you do, you'll reach a room with a lever that you can pull. Do so and you'll ring the second bell, which triggers a scene showing a giant beast raising the portcullis that you may remember finding at the the end of a forest path near the blacksmith at the Undead Parish. That opening now leads to a new area known as Sen's Fortress, though you probably shouldn't head there just yet.
Once that scene showing the rising portcullis ends, head around behind where you pulled the lever and you can descend a staircase if you like. That will take you toward a new area known as the Demon Ruins, but you're not necessarily ready to venture there just yet. However, along the way you'll find a room with a large circular plate on the floor.
Near the plate, look at the path descending ahead of you. Then roll against the wall to the right. Part of it will dissolve to reveal a wide, hidden path. You can head along that path to meet with a character. Talk to him and answer “Yes” to his question to be allowed through into the next chamber. There, you can rest at a bonfire and talk to a woman who extends partially from the wall to the right. You can join her covenant (Chaos) if you like.
If you join the Chaos covenant, you'll receive the Pyromancy: Great Chaos Fireball item but abandon your former covenant. By joining the new covenant, you'll be able to then talk to the nearby character and improve your Pyromancy Flame (at the cost of souls for each improvement) so that your magic of that sort does more damage. It's in your best interests to join this particular covenant if you are a character of that class, though you won't have that option otherwise.
When you're done there, you can leave and resume your adventure. You have a number of good options for how you wish to proceed, but perhaps your best bet is to head to the Catacombs area next. You may also have luck in Darkroot Garden, if you haven't already cleared that area and want to try switching up the order of things, or in The Great Hollow.