“Off-chain” refers to a transaction that happens outside of the main blockchain and isn’t published there. The most intuitive form of a second layer (or off-chain) transaction is for two parties to agree to a debt between them. If we establish that I owe you one Bitcoin, our agreement is a “transaction” that stays valid by virtue of the fact that we trust each other. There are much more structured versions of off-chain transactions however, such as Layer 2 solutions like the Lightning Network, or how you can trade Bitcoin on the 1Bitcoin exchange without needing that transaction to be written on the blockchain.