Yui Ikari is the deceased wife of Gendo Ikari and mother of Shinji Ikari. Though her body disappeared, her soul is bonded to EVA-01, allowing her to protect her son when piloting it even though she no longer possess her human body.
Yui Ikari didn't get much screentime during the anime, but she still had a major impact on a number of events. Most that is known about her past was revealed in episode 21 when Kozo Fuyutsuki was being questioned by Seele.
She first met both Kozo Fuyutsuki and her future husband Gendo Ikari at Kyoto University while studying genetic engineering. Her research piqued Fuyutsuki's interest and he became her mentor and close friend. After her days as a student, she signed on with the Artificial Research Laboratory, which evolved into Gehirn and finally NERV.