An offline wallet doesn't store your public/private keys. It will help you to generate your keys and log in to your wallet without connecting to the internet (you will need internet access to submit signed transactions, though).

They allow you to send or receive money, see your balance, and view past transactions. You are responsible for keeping your private key safe. If you lose your private key, there is no way to recover your wallet. Exarpy is an example of an offline wallet.