Do bosses respawn in Dark Souls?
Apparently all ordinary enemies respawn, but so far I haven't encountered a boss that respawns, mini-bosses included.
In conjunction with zero298's answer, a "boss" enemy is any enemy that has a static, on-screen health bar. These bosses will only spawn once per new game. Only entering a new game+ will allow them to respawn. However, if you wish to face them again before entering ng+, you can use the White Sign Soapstone and assist other players in defeating them.
These enemies are: (in general order of encounter)
Area Bosses: Once these bosses are defeated, you cannot summon/be invaded in this area anymore.
- Asylum Demon
- Taurus Demon
- Bell Gargoyles
- Capra Demon
- Gaping Dragon
- Chaos Witch Quelaag
- Iron Golem
- Dragon Slayer Ornstein & Executioner Smough
- Cross-Breed Priscilla
- Ceaseless Discharge
- Demon Firesage
- Centipede Demon
- Bed of Chaos
- Seath the Scaleless
- Sif, The Great Grey Wolf
- The Four Kings
- Pinwheel
- Gravelord Nito
- Gwyn, Lord of Cinder
Optional Bosses: defeating these do not affect summoning/invasions (with the exception of the Undead Asylum; since you have already defeated the area boss before invasions/summoning became available). This means if you wish to summon help for these fights, you will need to do so before fighting the main boss.
- Stray Demon
- Moonlight Butterfly
- Dark Sun Gwyndolin
DLC Bosses:
- Sanctuary Guardian
- Knight Artorias
- Manus, Father of the Abyss
- Black Dragon Kalameet (Kalameet is on optional Boss for the DLC)
There are also a series of enemies throughout the game that are not really considered "bosses" (sometimes referred to as "mini-bosses"), that will only spawn once per new game:
- Black Knights (they do re-spawn in the Kiln of the First Flame)
- Hellkite Dragon
- Hydra (Darkroot Basin/Ash Lake)
- Havel the Rock
- Armoured Tusk
- Butcher
- Golden Crystal Golem
- Giant Undead Rat
- Parasitic Wall Hugger
- Undead Dragon
- Undead Prince Ricard
- Prowling Demon
Bosses will respawn after you beat the game and move to New Game+. They will of course also be stronger. So will all enemies.
If you just want to fight the boss again, you could lay down your soap sign and wait to get summoned. You will not receive another boss soul on victory though.