Billions of years ago, a hyperadvanced humanoid alien race created seeds of life and housed these in seven 'moons', containers designed to transport them out into space so they can seed other worlds with life.
Some seeds are 'fruits of life', bringing forth immensely strong and hardy creatures. The other seeds are 'fruits of knowledge', bringing forth creatures capable of developing technology and civilization.
The aliens decided that no single planet should ever get two seeds, because a mixture of the seeds of life and knowledge would result in creatures that are both physically powerful and intelligent. They would be too similar to the aliens themselves. Unacceptable.
Because the universe doesn't care about any of that ****, two seeds landed on Earth. The seed of life brought forth Adam, and the seed of knowledge brought forth Lilith.
The Spears of Longinus are devices included in the moons, which are supposed to prevent two seeds from germinating on the same world. The moon carrying Lilith arrived on earth second, but its spear was destroyed in the impact. This caused Adam's spear to trigger, rendering him dormant. Lilith did what she was meant to do, and spread primordial ooze all over the world, eventually giving rise to intelligent life: humanity, also known as the 'Lilin' due to their origin.
Humanity eventually find the dormant Adam and run all sorts of experiments on him. They even end up inserting human DNA into him, triggering the 'Second Impact', a near-apocalypse.
Humanity survives, but just barely. Then, just over a decade after the Second Impact, the Angels appear. They're Adam's children in the same sense that we're Lilith's children, even though no one knows how or when Adam brought them forth.
Their sole motivation or instinct is to merge with Adam's remains, which are still in human hands, in order to cause a proper apocalypse and reclaim the Earth from Lilith's children, the second-comers who shouldn't even be here.
However, in the decade after the Second Impact, humanity has done all sorts of weird ****. Using tissue from Adam and Lilith, they've grown and engineered massive fighting cyborgs named Evangelions. They're biological creatures at their core, but they're been fitted with all sorts of sci-fi tech in order to keep them under control. This sort of thing always works out for the best and never-ever goes wrong in sci-fi.
Angels and Evangelions duke it out in classic mecha vs alien fight scenes, a Lilith-Adam crossbreed angel named Kaworu infiltrates humanity's Evangelion defense force, and everything goes tits-up.
Specifics about the ending vary wildly based on what version of the story you're watching, but they're all pretty bad endings as far as life on earth is concerned.
Also, there's a lot of pop psychology and pop philosophy mixed in. Some characters suffer all sorts of psychological trauma for added drama, and certain philosophical concepts are name-dropped for added 'depth'.
There's a lot of talk about souls, but you need to keep in mind Evangelion has chosen to define souls as magical/sci-fi forcefields. It's technobabble, rather than serious philosophy.
Still, pretty decent show about fighting robots with a 'hidden' backstory about life originating from space.