There was an ancient god-like race who created two kinds of Eggs, Black and White, each imbued with half of their attributes, in order to spread life across the galaxy. Each contained a progenitor entity which would spawn similar beings; in Earth's case these were Adam (Fruit of Life/Angels) and Lillith (Fruit of Knowledge/Humans).
No planet was supposed to be seeded twice, but a Knowledge Egg crash-landed on Earth which already had a Life Egg (First Impact). The collision destroyed Lillith's control device, the Lance of Longinus, allowing it to suppress Adam since Adam's Lance was still intact. It bled LCL (El Ciel) from itself, a mass of orange goo which could divide itself into multiple consciousnesses and which came to make up all life on Earth, aka "Lilim" (everyone is depressed because they subconsciously want to return to being whole but their AT Field, aka ego, drives others away). The Egg itself later became known as the Geofront.
Uniting Life and Knowledge would result in a god-like being similar to the creators of the Eggs, and just about every faction is trying to do this in order to achieve their goals; Eva-01 was needed for Instrumentality because it was grown from Lillith while the other Evas were grown from Adam, though I'm not sure if the Angels were aware of this. Yui let herself be absorbed into EVA-01 for unexplained reasons but Gendo can't stand being away from her and wants to hijack everyones' plans to bring her back. Lillith wants to reabsorb the Lilim so she can fight Adam at full strength. Rei takes pity on Shinji (at least the last clone does) and wants to give him a chance not to be a puppet.
Second Impact was the result of an experiment where SEELE tried to use the Lance to control Adam. Some really weird stuff happened, including IIRC some human DNA becoming sentient, diving into Adam and becoming Kaworu.
Oh, and the soul in EVA-02 isn't Asuka's mother, just the part of her that loves Asuka.