The Near Third Impact is a near cataclysmic event which occurred at the end of Evangelion: 2.0 You Can (Not) Advance and caused the Third Impact in the Rebuild continuity. The aftermath of the Near Third Impact is explored in Evangelion: 3.0 You Can (Not) Redo, with the effects being far more widespread than it appeared in Evangelion: 2.0.
In Evangelion: 2.0 You Can (Not) Advance, Shinji inadvertently initiates Third Impact during his fight with the 10th Angel. Having learned that Rei had been assimilated by the Angel, Shinji became determined to rescue her, no matter the cost. His overwhelming desire caused Unit-01 to begin transcending its boundaries, allowing it to reactivate (despite being completely out of power) and display strange new abilities that quickly incapacitated the Tenth Angel. As Shinji attempted to draw Rei out of the Angel's Core, Unit-01's transformation accelerated, exposing its own Core and manifesting a Halo.
When Shinji successfully pulled Rei from the Angel's remains, the latter manifests as a giant version of herself made of the Angel's blood and fuses with Unit-01. At this point, the Evangelion unit was at the center of a massive vortex and a cross-shaped energy wave, with a third eye on its head and several Wings of Light protruding from its body. However, Near Third Impact was averted when Kaworu Nagisa arrived in Evangelion Mark.06 and impaled Unit-01 with a massive lance, immobilizing and seemingly deactivating it possibly to cause the Third Impact.
In Evangelion 3.0 Kaworu shows Shinji the result of the Third Impact, which happened after the Near Third Impact. The ruins of Tokyo-3 and the land surrounding it have been devastated by impact craters and upheaval, and everything is coated with a blood-red substance. Massive red cross structures dot the surface, and inert red Eva-like entities are littered throughout the area, in many cases melded together in various formations. Enormous fissures also scar the land, one of which resembles a kilometers-long mouth complete with giant teeth.
In addition, Third Impact or a related cataclysm had a dramatic effect on the Moon. Its orbit has been altered, bringing it much closer to Earth, and it is no longer tidally-locked with Earth. The Moon also seems to have gained an atmosphere of some kind, as cloud formations are visible. Lastly, red grid lines and splatters cover the Moon's surface (some of the splatters were present previously, as evidenced by views of Tabgha Base, although a full view of the Moon seen in the post-credits scene of Evangelion: 2.0 You Can (Not) Advance seemingly indicates that those splatters were not large enough to be visible from Earth which might indicate that they were caused by the Third Impact that happened after the Near Third Impact).
Kaworu states that these are the product of mankind attempting artificial evolution. The blood stained Earth itself has become uninhabitable for humans as a result. Ritsuko's words imply that if the Third Impact had continued, the event would have exterminated all life on Earth.
In Evangelion: 3.0+1.0 Thrice Upon a Time, it is shown that Ryoji Kaji sacrificed himself to stop the Third Impact which was caused by the activation of Evangelion Unit 01 opening The Doors of Guf, also known as the Near Third Impact.
Relation to the Human Instrumentality Project
The Third Impact is required for Gendo's scenario for the project. SEELE did not want the Third Impact to happen as it would not go according to their plan so it is likely SEELE knew the Third Impact would begin. This is the probable reason for the sortie of Evangelion Mark.06. The other probable reason Mark.06 was sortied was to resurrect Lilith and have their version of the project come to be. It seems their plan for the project failed as Lilith was depicted as decapitated in Evangelion: 3.0 You Can (Not) Redo. So it would seem the Fourth Impact was a back up plan for SEELE as Asuka stated "Shinji was SEELE's trump card".