Tokyo-3 is the fortress-city where Nerv Headquarters is located, meant to serve as a bastion against attacks by the Angels. Nerv's Evangelion Units are launched from the city to fight the Angels, and Nerv's Evangelion facilities and weapons caches are located in the city. Tokyo-3 was built in the location of the pre-Second Impact city Hakone, in Kanagawa prefecture, on the northern shore of Lake Ashinoko.
Tokyo-3's buildings are capable of partially retracting themselves beneath the surface of the Earth for defense. Beneath Tokyo-3 is a massive subterranean cavern termed a "GeoFront" (which is secretly not really a natural formation, but the Black Moon of Lilith).
Officially, Tokyo-3 was built to eventually surpass Tokyo-2 as the capital of Japan, but its true purpose is to coordinate the interception of Angels headed for Japan (and around the world).
Tokyo-3 spans roughly 1.5 km in length from one end of the city to the other, arranged in 5 "blocks", with each "block" section being 300 meters long.
Tokyo-3 was damaged(to unknown but probably limited extent) by 3rd and 4th Angels Sachiel and Shamshel, and heavily damaged by the attack of the 14th Angel Zeruel, though repairs were subsequently made. However, not long after this in order to defeat the 16th Angel Armisael, Rei Ayanami self-destructed Evangelion Unit-00, resulting in a massive explosion which destroyed a large swatch of the city, creating a new artificial lake in the blast crater (though the section containing Misato Katsuragi and Rei Ayanami's apartments survived).
During The End of Evangelion, Seele working through the UN orders the JSSDF to make an all-out attack on Tokyo-3, culminating in the JSSDF dropping an N² Mine on Tokyo-3, vaporizing the entire city and leaving the GeoFront below open to the sky. For the moment, however, Nerv HQ survived because it was located far beneath the city. However, Third Impact soon began, and the combined Rei/Adam/Lilith hyper-being raised the Black Moon out of the Earth in its massive hands, suspending it thousands of feet in the air, until the Black Moon exploded, destroying the last trace of the Tokyo-3 complex.