[bnegreve]

A good definition from the PDF:

He wanted, in fact, to show that it is possible to convince without revealing, and so without unveiling his secret.

Or from wikipedia [1]:

In cryptography, a zero-knowledge proof or zero-knowledge protocol is a method by which one party (the prover) can prove to another party (the verifier) that a given statement is true, without conveying any information apart from the fact that the statement is indeed true.

[1] http://en.wikipedia.org/wiki/Zero-knowledge_proof

[ecesena]

I think the privacy component is also important (somehow hidden in "without unveiling his secret", but not so clear imo). When Alice proves her knowledge to Bob, no external party would "believe in the proof", as A & B might have colluded.