Short answers, pulled from the story.
Capcom released Darkstalkers: The Night Warriors in 1994. This title shifted the video game landscape by introducing a gallery of monsters drawn from international folklore instead of human fighters.
Pyron is the alien overlord who seeks to conquer Earth in the Darkstalkers story. His arrival forces the Darkstalkers out of hiding to oppose his rule.
Lilith is a fragment of Morrigan Aensland's soul that was separated at birth to keep her power in check. She later seeks to reunite with Morrigan to take control of her body.
Madhouse produced the anime miniseries Night Warriors: Darkstalkers' Revenge. The series was released in 1997 and stayed faithful to the characters' in-game personalities.
Jedah Dohma seeks to combine all the souls of the Darkstalkers into one singular being to save the realm of Makai. His plan involves luring other characters into a trap to use their collective souls to remake the realm.