On-chain NFT
An on-chain token means the art is stored on the blockchain just like the token and as long as the blockchain (node) exist, so too will the art remain indefinitely.
Off-Chain NFT
An off-chain token can be stored in one of two ways. The normal internet where you will be directed to a storage website via a link (metadata) attached to your token, or a safer practice of P2P, created to work alongside the blockchain to help project store images off-chain but in a secure way.