Bosses

Major Bosses
These major bosses can only be killed once in your game and will restore your humanity once killed. The required bosses must be killed to finish the game, the optional ones can be killed, but be aware this may have some negative consequences.

Asylum
You can kill him the first time you meet him, but it's very hard and tedious. If you just ignore him and get to the door on the left side of the room, you'll finally get your weapon, spells and shield then you'll be able to kill this Demon and get the Big Pilgrim's key. Although, on a NG+, you'll start with all your weapon and equipment, allowing you to kill him during your first encounter.
 * Asylum Demon (Required) but also Optional the very first time you meet him in a New Game.

Asylum Revisited

 * Stray Demon (Optional)

Undead Burg
Blighttown can be access via Undead Parish -> Darkroot Garden -> Darkroot Basin -> Valley of Drakes -> Blighttown without needing any keys and only having to kill the Taurus Demon.
 * Taurus Demon (Required)
 * Capra Demon (Optional) You do not need to kill him to get to the second bell

Undead Parish

 * Belfry Gargoyles (Required)

Depths
Blighttown can be access via Undead Parish -> Darkroot Garden -> Darkroot Basin -> Valley of Drakes -> Blighttown without needing any keys and only having to kill the Taurus Demon.
 * Gaping Dragon (Optional) You do not need to kill him to get to the second bell.

Blighttown

 * Chaos Witch Quelaag (Required)

Darkroot Garden / Basin

 * Moonlight Butterfly (Optional)
 * Great Gray Wolf Sif (Required)

Sen's Fortress

 * Iron Golem (Required)

Anor Londo

 * Dark Sun Gwyndolin (Optional, breaks covenant Blade Of The Darkmoon)
 * Dragon Slayer Ornstein / Executioner Smough (Required)

Painted World of Ariamis

 * Priscilla the Crossbreed (Optional, not hostile until attacked)

Dukes Archive / Crystal Caverns

 * Seath the Scaleless (Required)

The Catacombs

 * Pinwheel (Required)

Tomb of the Giants

 * Gravelord Nito (Required, breaks Covenant Gravelord Servant)

The Abyss

 * The Four Kings (Required, Covenant of Artorias ring needed)

Demon Ruins
Shortcut to Lost Izalith
 * Ceaseless Discharge (Required)
 * Demon Firesage (Optional if you are in the Chaos Servant Covenant (rank +2) and use the Shortcut from Lost Izalith to Demon Ruins)
 * Centipede Demon (Optional if you are in the Chaos Servant Covenant (rank +2) and use the Shortcut from Lost Izalith to Demon Ruins)

You'll find the shortcut to Lost Izalith in Demon Ruins just before the room where you fight the Boss Demon Firesage.

As you face that room, look to the right, and you'll see a hole in the wall. Use the tree roots to reach the ground floor and then go to the end of the long hall where you'll find a locked door. The way to unlock that door is to join the Covenant of the Chaos Servant, and then level that Covent to +2 status. That will require 30 Humanity to raise the Covenant that high. After you raise the Covenant to +2, then you can touch the locked door and it will open.

Lost Izalith

 * The Bed of Chaos (Required)

Kiln of the First Flame

 * Gwyn, Lord of Cinder (Required)

Mini-Bosses
These are stronger enemies considered as Mini-Bosses, cause each of their iterations (except for Black Knights later on in the Kiln of the First Flame) only spawn once a game.
 * Black Knight
 * Armored Tusk
 * Titanite Demon
 * Havel The Rock
 * The Butcher
 * Giant Rat
 * Hellkite Dragon
 * Hydra
 * Zombie Dragon
 * Golden Crystal Golem
 * Parasitic Wall Hugger

Order of major bosses
Since Dark Souls is more-or-less an open world RPG (though "seamless" would be the better word), it's not always easy to find your next target/boss you should head for advancing the story. So this is a list that shows an example of an order in which you can defeat the bosses. But since you can access a lot of places right from the start there are not really many rules for an order to defeat the bosses, except for e.g. the four lords of chaos whose souls you need to fill the Lordvessel, as those can only be defeated after the orange fog has been lifted by Gwynevere, Princess of Sunlight.

Ringing the first bell

 * You start off in the Ayslum, the tutorial area. The boss here is the Asylum Demon, but the very first time you meet him it's optional, and you propably won't be able to beat it yet (except for the strategies noted on the boss page), so you can skip it when you start a New Game (on a NG+, you'll start with all your weapon and equipment, allowing you to kill him the first time you meet him).
 * Your next boss encounter (which you can't miss, you wont' be able to bypass it) is the Taurus Demon in the Undead Burg. You'll find it after the first bonfire between two castle towers ontop of the castle walls.
 * Venturing on you'll have to fight the two Belfry Gargoyles on a rooftop. They're guarding the first bell you need to ring.

Ringing the second bell

 * Next step is the Capra Demon far down in the depths of the Undead Burg. Kill it and you'll get a key to open up the door that leads to the next area.
 * Now it's time to find your way to the second bell. Your first boss on this journey will be the Gaping Dragon, located in The Depths. This area is accessed via the Undead Parish.
 * From the depths you can now access Blighttown with a key, there is no required boss here, but the optional mini boss Parasitic Wall Hugger. Only kill it if you need the items below it or want it's souls. It's an easy battle anyway (especially ranged)
 * Once you made all your way down to the swampy bottom of Blighttown walk up to that spiked hill you can see. It contains the next imporant boss, the Chaos Witch Quelaag. She is guarding the second bell.

Meeting some more requirements
Note : These two below are required, but you can do them in any order you want.
 * Next boss is one of the easiest in the game, namely the Moonlight Butterfly, in the Darkroot garden.
 * Great Gray Wolf Sif, located in the Darkroot garden. He drops the Covenant of Artorias, a ring need to traverse the abyss, you need this ring to fight The Four Kings later on.
 * Pinwheel, located in The Catacombs. He is on the way to Gravelord Nito, so you have to kill him anway, giving you the ability to enlighten the flame. Although, you won't be able to progress further into the Tomb of Giants afterwards since you need the Lordvessel to unlock the huge yellow fog, leading to Gravelord Nito.

Getting the Lordvessel
The Lordvessel is a very important item in the story of Dark Souls that you need to get your hands on. Getting it will also lift the orange fog opening up new areas you couldn't access until now.
 * Ringing the second bell made a huge colossus open up the gate's to Sen's Fortress, a nasty dungeon full of traps and enemies. Make your way up to the top and fight the Iron Golem. It's a pretty easy battle compared to the nasty dungeon you had to go through. Don't forget to activate the lift here for a shortcut.
 * Once the iron golem is history, touch the ring of light at the middle of the boss area, and you will be dragged away to Anor Londo (talk to the gargoyle atop the wall to return if you need to). Your next target (which may take time reaching) is to kill the dynamic duo of Dragon Slayer Ornstein / Executioner Smough. One of the harder battles, but never forget you can summon other players for help.
 * Now the path (elevators left or right) to Gwynevere, Princess of Sunlight is opened up. Talk to her and get the Lordvessel. Also the orange fog is lifted. Your next step is to fill the Lordvessel with the four powerful souls of the chaos lords.

Filling the Lordvessel (Killing the four lords of chaos)
So after the orange fog is lifted, four new areas open up, each with one of the chaos lords as their bosses. Note :There is no order in which you have to kill them. You just need to retrieve their souls and put them in the Lordvessel, so select as you like from below :
 * Seath the Scaleless, the chaos lord boss residing in the Crystal Cave, the are you reach through The Duke's Archives (accessed via Anor Londo form the first bonfire to the left).
 * Gravelord Nito, the chaos lord guarding the Tomb of the Giants (accessed via The Catacombs through the boss area) . Note that you may want to find yourself the Skull Lantern, or you won't be able to see a thing. There are two bonfires here, and from the second one it's only a few steps to Nito.
 * The Bed of Chaos, the chaos lord in Lost Izalith. You can get there through the Demon Ruins, very quickly if you have activated the bonfire at Quelaags Sister. There is only one bonfire in this area, but the bossfight is extremly simple as you don't fight the boss itself but rather destroy three parts of it. Note that you have to defeat several other bosses first to get there, namely the Ceaseless Discharge to open up access to the next area, the Centipede Demon to get the lava ring (needed for Lost Izalith) and the Demon Firesage.
 * The Four Kings, they are located in The Abyss. To access them go to the New Londo ruins, talk to Ingward and he'll hand you a key to a lever that drains the water. Now you can go to a tower that leads down to these bosses. Equip the Covenant of Artorias and jump into the hole to reach them.

Confronting Gwyn
Important Note : After killing Gwyn, the game is over. If you touch the bonfire or leave the game, it's over and you'll see yourself starting from the beginning with a New Game Plus. So if you want to do other things (optional bosses, farming, etc.). Don't kill him yet!
 * Now return to the Firelink Shrine and go down with the help of Kingseeker Frampt and put the souls in the Lordvessel.
 * The door behind it will open up and you can get to the final boss fight of the game against Gwyn, Lord of Cinder.