Mike put it about as plain as it can be. It is simply the negative form. One blatantly open point is that you don't end with ず (unless your purposefully ending your sentence prematurely)
Like Mike's example.
お金を忘れずに「何何します」
手を洗わずにご飯食べないで
Ooh -- this one is interesting. You could say:
手を洗わないでご飯を食べないで but you could not say:
手を洗わないでご飯を食べず「に」
It's a negative conjugation of the verb but is used to connect, and needs something to follow it. That's the key.
You can also use it as an adjective, as in your first example.
帰らずの道 = The road with no return (no turning back from here)