The apparent "enemies" of Neon Genesis Evangelion are the mysterious beings known as Shito, or Angels. As foretold in the Dead Sea Scrolls, the seventeen Shito are prophesized as being the final trials that mankind must endure before attaining its final evolution.
Originally, there was only one divine, godlike being known as Lilith. But the First Impact split this being into the first two known Shito, Adam & Lilith. It was from Adam that the other fifteen "lesser" Shito emerged and it is to him that they must return (merging with Lilith would set off the Third Impact). Note that this is why they "attack" Tokyo-3; both to merge with Adam (even though the being in NERV HQ is actually Lilith) and to test man's willingness to evolve.
The reason the Shito attack Tokyo-3 even though Lilith's there and not Adam is because Adam was reduced to embryo form and thus is not as easily located by the lesser Angels as Lilith. Add to this that Lilith's energy and Adam's energy are very similar, making it very easy for the Angels to confuse the two, like Tabris, the Seventeenth Angel had done.
One of the most interesting qualities of the Shito is that each one is unique. In fact, aside from their AT Fields, incredible strength, and need to reunite with Adam, the Shito really have practically nothing in common whatsoever! Their size ranges from being as small & microscopic as a virus to as large as a skyscraper. The appearance of each Shito is also unique, being anywhere from resembling an insect, to being as simple as a cube or sphere, to even taking the shape of a human! Their abilities range as well; some Shito are armed to the teeth while others have nothing more than their AT Fields and simple brute strength. It is this individuality that makes every battle in Evangelion different and interesting.
Oh, and for those who are wondering...
AT Field - AT Fields, or Absolute Terror Fields, are nearly impenetrable barriers that surrond the bodies of Shito. Also called "Light of Mind Barriers", it is said that anything within an AT Field is Holy Territory and cannot be touched by sinful beings (one of the reasons humans cannot break one through conventional means). Believe it or not, humans also possess AT Fields. But the barriers around humans are much more mental than physical, and it is these individual fields that act to keep man from uniting as the Lilim once more.
I - Lilith
Evangelion: Death & End of Evangelion © Studio Gainax. Licensed by Manga Entertainment, Inc. All Rights Reserved, or as otherwise directed by Manga Entertainment. Created by Hideaki Anno.