Third Impact
Third Impact (サードインパクト) is the event, as predicted by the Dead Sea Scrolls in the possession of Seele, which will usher in a new stage of evolution and existence. Third Impact is a prerequisite for the Human Instrumentality Project.
The event is termed "Third Impact" in reference to "Second Impact", when Adam was awakened by the Katsuragi Expedition funded by Seele. "Second Impact" itself was so termed due to the ruse that it was a meteorite impact.
There are actually several alternate versions of "Third Impact", with different outcomes. Also, some versions can be achieved through several alternate scenarios. Roughly, the potential versions of Third Impact which could occur are:
The Angels' version
Achieved by an Angel reuniting with Adam, being held in Tokyo-3. Unlike the various versions which different humans are trying to achieve, the Angel-version of "Third Impact" does not involve "Instrumentality": trying to forcibly re-unite Adam and Lilith for humanity to achieve a new evolutionary level of existence. The Angels want to reunite with Adam, perhaps as a coherent motive or maybe as simply an instinctive drive.
Like what happened during Second Impact where all life was erased in Antarctica (aside from Misato), presumably, an Angel reuniting with Adam will result in Adam generating a world-wide Anti A.T. Field, eliminating all Lilith-based life on the planet, from bacteria to humans. The result of Second Impact was what Fuyutsuki describes in Episode 12 as: "The South Pole, a world of death which permits no life to exist." Kaworu, in Episode 24, also mentions that "Because it's my destiny to continue to live even if it may result in the destruction of humanity", implying that his goal to reunite with Adam would cause the end of humanity.
Seele's version
Seele wanted to initiate Third Impact to bring about Instrumentality. They would use an Anti A.T. Field to neutralize the A.T. Fields that separate human beings from each other, causing all of humanity to revert into a giant ocean of LCL, freeing their souls. All the souls of Earth would then be collected inside the Adam/Lilith hybrid being and her Egg. This could be achieved through several alternate scenarios involving the Mass Production Evangelions, the Spear of Longinus, Lilith or Evangelion Unit-01 as Lilith's substitute.
Gendo Ikari's version
Gendo Ikari wanted to reunite with his wife Yui Ikari, whose soul was trapped inside of Evangelion Unit-01. It is not entirely clear what the full scope of Gendo's version of events would entail for the rest of the human race (in all probability, all he cared about was Yui). However, judging from Gendo's remark during his conversation with Seele at the beginning of The End of Evangelion that "destruction creates nothing", it is possible that Gendo's version of Third Impact would not have been as destructive for the human race as a whole as Seele's version. As Gendo's full plan was ultimately not realized, it can not be determined with certainty exactly what would have happened. However, it is clear that Gendo's version of Third Impact involved the use of Rei Ayanami (a clone of Yui Ikari serving as a vessel for the soul of Lilith), Lilith, and Adam (which he had fused to his hand). It is assumed that this fusion would allow him to reunite with Yui again inside Eva-01.
Actual Third Impact
Third Impact occurs during the climax of The End of Evangelion. At the beginning of Episode 26', the Mass Produced Evangelions capture Shinji along with Eva-01. Along with the Spear of Longinus, a ritual is performed that creates an inverted Tree of Sephiroth in the sky, and turns Eva-01 into the Tree of Life. The Mass Production Evas unleash their A.T. Fields, causing a giant explosion around the GeoFront. In its wake, Lilith's Black Moon is revealed.
Meanwhile, in Terminal Dogma, Gendo attempts to fuse the Adam embryo in his right hand with Rei. Rei rejects Gendo, taking his hand along with Adam's embryo, and merges with Lilith, thus achieving the "forbidden union of Adam and Lilith".
After assimilating Adam, Lilith begins to merge with Eva-01. During this process, Shinji's states, "Nobody cares whether I exist or not... Nothing will change. So, everybody just die." This wish seems to manifest itself into reality, as Adam/Lilith proceeds to create an Anti A.T. Field which spreads across the globe, reverting all Lilith-based lifeforms into LCL. The liberated souls of humanity are gathered by Lilith and funneled into the Black Moon (her "Egg"). When this process is completed, Instrumentality begins.
This is clearly not what Gendo had intended, but the Seele council seemed fairly content with the outcome (even if it had not happened quite as they had planned). However, Shinji Ikari ultimately rejects Instrumentality, and the Black Moon is destroyed, dissipating the souls back across the Earth and into the oceans. Lilith's body falls apart and descends to Earth, and she apparently dies.
All disembodied individuals are left with the option of recreating their bodies and returning to a life in the physical world, regardless of the pain it would inevitably entail. Shinji, and later Asuka, return to the vicinity of Hakone, but the fate of everyone else is left open.
Rebuild of Evangelion
In Evangelion 2.0, Eva-01 enters a state of abnormal power during the battle with Zeruel, evidently far beyond the "normal" berserk mode, demonstrating such capabilities as an A.T. Field arm, optical beams and an angelic halo above its head. Shinji and Eva-01 proceed to rip Zeruel's body open, following which Shinji "dives into" a dark void within the Angel to rescue Rei, with the remains of Zeruel and Eva-00 turning into a giant red Rei that fuses with Eva-01. During this process, the halo above Eva-01's head turns into what looks like a black hole, similar to the formation above the site of Second Impact. The wrecked remains of Tokyo-3 are utterly destroyed, and Eva-01 is set to transform into a new lifeform. Ritsuko and Misato recognize the scene as Third Impact. The event is stopped by Kaworu in Eva-06. There is little explanation as to the nature of the event or how it came about in such a fashion, but it is implied that had Kaworu not stopped it from proceeding, Third Impact would have wiped out all life on Earth and elevated Eva-01 to godlike status.
