Creatures form a set of diverse character types in American Horror Story. Most of those depicted are humans that have been modified (either while living, or crafted from cadaver parts) through magic, science, or an unknown alchemical process with elements of each.
In "Murder House"
Infantata is a crafted entity, created by Dr. Charles Montgomery from the remains of his murdered son Thaddeus in Murder House. He crafted other chimera as well in a form of crypto-taxidermy, but there is no evidence that any of these attained any consciousness.
Bastien, a house slave of Delphine LaLaurie, was tortured and mutilated by her. This culminated in her execution of him by placing a severed bull's head over his own, as she rejoiced in having made her own Minotaur.  He was reanimated by his lover, Marie Laveau.  Over time, the bull's head has integrated itself to his body and he developed additional bull-like characteristics. 
These corpses have been reanimated by magic. They are merely puppets of the ritual caster, such as those raised by Marie Laveau to exact vengeance on her enemies. The end result seems largely based on the intent of the caster and the rite used.
Please note that this is different from those who have been restored to life by the resurgence ability. Those affected are no longer dead and are alive, not undead.
Immortals are humans that have been fortified to be unaging, and eternally require no sustenance. They are not immune to injury, but their individual body parts will continue to survive once severed.  Marie Laveau gained immortality by making a pact with Papa Legba, which entailed her having to sacrifice an innocent soul every year at an allotted time.  Delphine LaLaurie was cursed with immortality when she drank a vial of Marie Laveau's tears - a punishment enacted by the Voodoo Queen for her torture of black slaves. 
Afflicted are humans who have been infected with an ancient blood virus. They display heightened physical characteristics, such as an extremely powerful immune system, a body that does not age, rapid healing and increased speed, at the cost of requiring a steady diet of human blood.
Despite being able to survive for decades without blood, food, or water, the Afflicted can still be killed by violent means. They are weakened by sunlight, and can suffer from certain diseases they held before becoming infected, particularly neurological disorders that the virus cannot destroy such as Alzheimer's Disease. They are also weakened by consuming blood polluted by drugs or disease.
Afflicted are not dead, though the virus is strong enough to save their lives when they are close to death, as is the case for many. Once killed, their souls are capable of becoming trapped as ghosts, just like any other human.
Despite popular culture depicting variations of them, the Afflicted exist in complete secrecy, with knowledge of the virus being sparse and spread out across time and different countries. Even the highest of witches do not seem to have any knowledge of their existence.
NOTE: The term "afflicted" is used as a placeholder here until a term is given inside the show's universe.