The Angels' version
Adam's wings expanding during Second Impact
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 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