grammar - The difference among ずに、ず、ないで、なくて

I know they are all the negative form of a verb, but I just don't know the difference. I still remember when I was learning Japanese, my teacher told me that ずに equals ないで in a more ancient way, but what about the other two? When should I use ないで, and when should I use なくて? And how about ず? Thank you.