"Off-chain," as opposed to "on-chain," is a technical term referring to blockchain technology. A piece of technology that is "off-chain" is something that affects how a blockchain manages storage space, processes transactions, or similar functions, but exists as an external piece of technology that does not directly influence the blockchain. In other words, an "off-chain" software function is something that "plugs into" a blockchain to improve its technical capabilities - sort of like how an external hard drive or USB stick improves the memory management capabilities of a desktop computer - as opposed to altering the structure and/or function of the blockchain itself. An example of off-chain technology is the Lightning Network.
Many cryptocurrency trading platforms conduct many of their transactions off-chain simply by marking their internal ledgers to show transactions between customers. In these cases, often involving retail customers, the customers' cryptocurrencies have been deposited with the platform prior to the transaction and are stored in the platform's wallet.
- Fraud proofs — Secure on-chain scalability. Hackernoon.