When humanity discovered the angel
Adam in the
Antarctic, the attempt to shrink him down to
embryo form caused a devastating release of energy, the
cover-up was that it was a large meteor impact, it was the second most destructive since an asteroid collided with
Earth 4 billion years ago (which resulted in a large chunk of the Earth to fly into space, where it became the Moon), thus was dubbed
Second Impact.
Few people in the world knew the true cause of
Second Impact, it is even possible that the
angels were in some way the cause of
First Impact too (the large cavern
Adam was discovered in was code named "white moon", it is possible that the moon is this white moon and linked to
Adam like how the
black moon is linked to
Lilith). What was also known by these people was yet another catastrophic event, which they referred to as Third Impact.
Third Impact would be caused when
Adam, the source of life, comes into contact with an
angel.
Kaworu said that only one life form can inherit the
Earth, Third Impact would mean salvation for the life form that merged, and the end for all other life.
The
Evangelions were created with the story of protecting
Earth against the
angels, beyond
SEELE few knew about what the angels really were, not even governments, so they provided the funding. The people working for
NERV, who knew about
Second Impact believed that they were working to stop Third Impact. It was, however,
SEELE's plan that the
Evangelions instigate Third Impact, only with their own scenario.
The
Human Complementation Project would evolve humanity far beyond what they are, to remove the
AT field between people so that all could become one, not only that but would elevate themselves to the level of
God.
Gendou Ikari had different plans for Third Impact, intending to use it in order to be reunited with his wife
Yui Ikari.
However the two plans conflicted and the result was that the direction of Third Impact was led by
Shinji Ikari.
SEELE had begun their plan, launching the
Mass Produced Evangelions to destroy the remaining
NERV units, killing
Asuka Langley Sohryu. Shinji intended to destroy
EVA-01 to stop Third Impact (it was an integral part of both plans), he tried to summon the
Lance of Longinus to destroy the Eva, but it stopped next to him, the mass produced Evas formed an image of the
Tree of Life focussing on Eva-01 and the lance.
SEELE started the ritual.
Rei ultimately rejected Gendou, she took
Adam and merged with
Lilith, thus Third Impact began, using the forbidden combination of
Adam and
Lilith. Rei/Lilith grew towards the Evas, and
Shinji Ikari. The
Lance of Longinus envelops
EVA-01's core and turns into the
Tree of Life. A large explosion extending from Japan released the
Geofront which in reality was the egg of
Lilith, the
black moon. Shinji is frightened and screams, when Rei/Lilith changes her face to show that of
Kaworu, this calms Shinji and he starts being asked questions about what he wants. Shinji has been chosen to decide the fate of humanity (it is unknown if it is Rei, Lilith, Yui or Kaworu asking these questions, if it is a mix of them, or if it is something else but the point of the questions is to determine what Shinji wants, what he thinks is best for himself will decide the fate of humanity).
Shinji has a lot of things going on in his head, trying to judge what he thinks of others, himself, what he wants. In the end when he realises the pain of the real world he decides that everyone should die. At this point the anti-AT field emitted by Rei/Lilith and the EVAs expands, covering the world. Everyone's
AT field is destroyed, their bodies turn into
LCL and their souls float up to the
black moon and into Lilith/Rei.
Shinji goes through a lot of philosophical questions, he sees different possibilities for worlds, then he is in the sea of LCL, where the
AT field doesn't exist, where you can't tell one person from another, the world without the thing that caused him pain, brings him true loneliness, it is a dead world. It isn't what he thought it would be, but for people to be separate again means restoring the
AT field, returning the fear of other people, of not knowing them. He understands that even after running away, from his father, from
NERV, from people themselves, things don't get better. He understands that though the
AT field can bring
hate,
fear and
loneliness, it can also bring
love, without it, there is nothing.
With his decision to restore the
AT field the giant Rei/Lilith falls apart, the souls of humanity spill out orbiting the Earth, the LCL floods into the seas, the EVA series die and fall to Earth.
People can return if they have the will to return.
Third Impact occurred and failed to produce the plans envisioned by Gendou or SEELE, leaving humanity left with a
post apocalyptic world. However, it did provide humanity with an
Ark to survive the catastrophe that would otherwise have destroyed them all.
Note: While there is a lot of what happened in End of Eva, it is not supposed to explain a lot of the other events and things that happen, it is just a description about Third Impact itself,
The End of Evangelion is the place for a full write up about the movie.