The First Impact is the meteor impact that occurred around 4 billion years ago which caused the Moon to split from the Earth, according to the giant impact theory, shown in a document in the seventh episode of the series. It is speculated that the "meteor" in question would be the Black Moon of Lilith, burying itself into the Earth by dislodging an equal amount of mass, thereby creating the White Moon, referred to as the Egg of Adam. The real life speculation is that a planetoid called Orpheus orbiting at a distance between that of Earth and Mars smashed into Earth I (the planet Earth before the First Impact), causing part of the smashed-together planets to separate and coalesce, eventually becoming the Moon. Some erroneously say that the First Impact was the Cretaceous-Tertiary extinction event, but there were no dinosaurs 4 billion years ago. Another erroneous speculation is that First Impact was God's destruction of mankind described in the Noah's Ark story, as this appears to fit more with the series in that all the other impacts destroyed most of Humanity, seemingly for its sins; but there were no humans 4 billion years ago.
The First Impact is never shown in the series or movies.
A cataclysm which followed the "contact experiment" conducted on Adam in the South Pole on September 13, 2000. The awakened and injured Angel created a strong anti-AT Field which caused an explosion that melted the Antarctic ice cap and caused a shift of Earth's axis; climate changes and gigantic floods followed. As a result, wars broke out due to scarce food and resources; these conflicts spread all over the world, and were ended on February 14, 2001 by signing what would be later called "The Valentine Treaty". By the end of these events, the human population of earth was halved.
Officially, there was no Adam and no "contact experiment". According to the cover-up, the Second Impact was caused by the impact of a meteorite that was too small (about 4 inches wide) and too fast (95% of the speed of light) to detect. A remarkably detailed description is shown briefly in the series in a text book, though it is highly unlikely it can be all read within the short period that it was exhibited.
An event precipitated when Ikari Gendou places Adam inside of Lilith, causing AT-fields to disappear and the subsequent merging of all souls into one. This takes place in the end of the series in the movie The End of Evangelion.