I am trying to get a mortgage through a program that loans us the money for closing costs and down payment. He informed us that we needed a "non responsible co signer".
What is that?

First if you don't have the money for closing or down payment you don't need to be buying a home, and you need to work to get good credit also before buying a home.

All co-signers are responsible for repayment of the debt if you don't that is what a co signer does, as for what they are not responsbile for is to ask the lender.

I am trying to get a mortgage through a program that loans us the money for closing costs and down payment. He informed us that we needed a "non responsible co signer".
What is that?
The cosigner is additional security, an additional payor, in case you default. He is saying that you (us) are not qualified at this point.