As in "

zero-

knowledge proof".

Let's say I need to prove to you that I know
some piece of information (example: the password
to a computer account, the location of your
kidnapped parents) - but I don't want to *reveal* the information to you. A zero-knowledge proof of
knowledge will allow me to do that. The phrase
"zero-knowledge" arises from the fact that I can
prove I know some information while providing you
with no knowledge about the actual content of the information in question.

Zero-knowledge proofs are typically statistical in
nature: you ask me a series of questions whose answers
depend my knowing the information in question; if
I can answer them all correctly, the chance that
I don't know the information in question becomes
negligible. Conversely, if I *don't* know the
information, my chance of answering all the questions
correctly is negligible.

Zero-knowledge proofs are covered in some detail
in Bruce Schneier's excellent book, Applied Cryptography.

I expect that Zero Knowledge Systems got their
name from this idea.