Lilith is a Caster-class Servant, a manifestation of the sister / first wife of Adam according to certain scriptures. She refused to submit to him, so she was cast aside and then Eve was created to take her place. As according to her lore, Adam and Eve are the genesis of humanity, Lilith is thus an Enemy of Humanity Servant, holding a deep-seated grudge against them all.
Though she is technically a succubus, she is also exceptionally faithful toward her God. However, her form of worship runs counter to anyone else's, as her doctrine is the Eden-type. Humanity as it currently exists is inherently heretical, and her goal is to reinstate God's will and bring back the sinless garden, Eden.
Her Noble Phantasm, Mistress of Eden, is a manifestation of that; as she did not participate in the Original Sin, but was one of Eden's residents, she subsequently became the only human able to call it home and didn't get kicked out. She is Eden's mistress, and she manifests it in different ways to serve her goals.
She has an extremely developed sense of smell, she picks up scents from a Conceptual level - when she refers to the "stench of humanity" she means it literally. Almost all humans reek of humanity to her, it's sickening and drives her disdain all the further. Some smell different than others, though; for example, in Crematory Quell (the story she's in), her Master's humanity "smells stale and muted" which lets her deal with it more easily, and her ally Berserker "smells so strongly of blood and avarice that it almost drowns out the humanity".
A very brief and abridged version of Lilith's lore.
