Eliza is Faust's first and only love. She died at 26 from a fatal illness she'd battled since childhood. While she was sick, Faust threw himself into his studies obsessively, determined to find a cure—and he succeeded. After she recovered, they married and opened a clinic together, and for a while they were genuinely happy.
That ended when a burglar broke into their home and shot her. She was alone in the clinic at the time.
Now Eliza's spirit serves as Faust's main spirit guide as he chases the title of Shaman King, hoping eventually to bring her back to life. As Faust gets stronger, she becomes more and more lifelike and present. Anna eventually helped summon her spirit back to Earth, which let Faust actually communicate with and see her again—a breakthrough that made his power surge since he now had a real human spirit rather than just a ghost. As the anime goes on, she gains more agency too, speaking more lines and showing more personality.
There's some speculation she might be named after the stage where The Tragical History of Doctor Faustus first premiered, though that's more fan theory than confirmed fact.
Content compiled by AnimeList.moe from publicly available sources.