A Party is a demon- specifically a Mud Beast- that has possessed a human body.
Parties are significantly more dangerous than a demon manifested into its true form, since a human vessel is more stable, with the brain allowing for more convoluted schemes. Just as a Hollow can, a Party is able to channel their powers through their body's flesh and blood. While Hollows must train to understand their own abilities, a Party only need only adjust to their acquired humanity in order to be effective. Parties are luckily very rare due to the Bahutha's relentless campaign to eliminate them, and the difficulty of possessing a human without killing it.
The only relevant Parties are patrons and employees of the Mystery Loft, a neutral ground where attacking humans is strictly off-limits. This hints at a larger culture between Parties, and that some Mud Beasts become relatively tame when granted an opportunity to express their humanity.
There is no evidence that a Lux or Apparition can possess a human. It is unknown if the original owner of a possessed body retains any awareness, as all currently practiced exorcisms result in death of the host.