I just finished watching the TV series. I haven't seen the movies yet. Here's my interpretation:
The central message of NGE is an anti-religious message: Don't base your morality on religion. Instead, think for yourself. Here are some of the symbols:
Eva = the religious-based morality that society expects from people.
Syncing with an Eva = conforming to that expectation. When Shinji has the 400% sync ratio, he conforms too much and becomes like an animal, losing his humanity.
Angel = Non-religious thoughts (which includes the natural morality of humans) or a person who has those thoughts.
NERV = religious extremist group that labels the Angels as evil, and wants to destroy the Angels and kill anyone who goes against NERV's beliefs. However, Angels are not really evil, and we never actually see them going around destroying cities intentionally (at least not that I can remember, correct me if I'm wrong). They only attack or defend against the Evas, and any destruction is a side effect of that fighting. Second Impact happened because, as Ritsuko states, "mankind found a god" and "tried to make the god his toy" (in other words, abusing religion and using it to justify immoral actions).
AT Field = a wall that encloses a person's mind (as Kaworu states). The Eva vs. Angel fights are like religious debates. The Eva tries to break through the Angel's AT Field to brainwash it into believing in religion, while the Angel tries to attack the Eva pilot's mind to show him that religion is wrong.
Evas were originally copied from angels = Religious morality is based on natural morality. All people have a natural sense of morality, but religious people lie to themselves that they need religion, possibly because they find it easier to understand (similarly, NERV pretends that Evas are better than Angels, and they use "restraints" to control the Eva's power to try to make it easier to understand).
Rei = typical mindless follower of religion who can't think for herself.
Shinji = very confused boy who has his own sense of morality, but he's insecure and worries that his morals are not good enough, so he's desperate for society's approval. He pilots the Eva not because it gives him purpose in life, but because he feels a moral obligation to save mankind. He tells himself it gives him purpose, but deep down he feels troubled and unfulfilled because he knows what he's doing is wrong. He becomes more suspicious when he is ordered to kill Toji. Toji's Eva does not actually malfunction. Commander Ikari discovers that Toji's beliefs are different from NERV's, and that is the real reason why he labels Toji an Angel and orders Shinji to kill him.
The final 2 episodes don't really happen. They are a hypothetical situation in which Shinji is able to step back and look at his own life (with some help). The purpose of these episodes is to show how his life could have been better if he wasn't so insecure. The scene with the empty world shows that if Shinji got his way and lived in his own perception of a perfect world, it would be an empty world, and not true happiness - this shows that Shinji is confused about what he wants out of life and has unrealistic desires.
Oh yeah, and the most important point... Kaworu likes Beethoven's 9th, and anyone who likes Beethoven's 9th must be good!