Warning Spoilers!

What is Shinji's link with Unit-01?

Fact
Yui, Shinji's mother is Unit-01. Well not exactly. In an experiment during Shinji's youth, Yui was conducting an experiment, supposedly she was the pilot of Unit-01 and she was dissolved into the unit like Shinji did in that one episode where he achieved 400% synch ratio. While he floating around as LCL, he encountered his mother inside, which was when she let him out...sorta. Also there is a scene near the end of the series, Gendou was standing right in front of Unit-01, and he said, "The time has come, Yui." This greatly explains Shinji's ability to synch without prior training with Eva, I think we all know episode 1!
Source: TV series

Reader Theories:

When I first started watching Neon Genesis I thought that maybe the plugs acted as the cerebral cortex for the EVAs. I mean, the spot they slip into reminds me of the spot you jab to pith a frog. However, the LCL fluid, the motherless kids, Yui showing-up to give Shinji a spark of energy when he was in the Angel Sphere ...

Gendo has been trying to get his wife back. He is trying to reawaken her by putting a child who is hungry for motherly comfort in a womb-type environment. When EVA 01 goes berserk you are looking at a pregnant female defending her unborn child!

Now, I haven't seen the movies nor have a read much about them yet. But I think Gendo has been using all of the resources, people, etc to get back the woman he loved and lost years ago.

-John Huber