Put some Bitcoins in a wallet, extract the secret key and write it on a piece of paper (tip: fold the paper to keep the key secret). You now have the ability to make an off-blockchain transaction with that piece of paper.
This is a good example of an off-chain transaction, however once you sell/give that paper to someone else that 2nd person would need to then transfer those coins to an address that they control as the other person (and maybe others) potentially have the private key.
A true off-chain transaction would be one that does not force the counterparty to transfer the coins right away.