Off-chain is explained alongside its opposite in On-Chain vs Off-Chain, which compares the two in detail. This page covers the word on its own and the many places it turns up.
"Off-chain" is an adjective for anything connected to a blockchain that is not itself written into the chain's blocks. Its partner term, on-chain, covers what is recorded on the shared ledger and verified by every node. The distinction matters because on-chain actions are public, final and paid for with network fees, while off-chain actions are usually faster and cheaper but rely on some other system, company or agreement for trust.
The word is applied to several different things. Off-chain transactions include transfers between customers inside an exchange and payments routed through the Lightning Network. Off-chain data includes prices and real-world facts that reach smart contracts through a blockchain oracle. Off-chain computation is work done elsewhere and then proven or settled on the main chain, as in many scaling designs grouped under the off-chain layer. There is also off-chain governance, the human discussion and coordination that decides how a protocol changes. In each case, the trade-off is the same: speed and privacy in exchange for less of the chain's built-in security.