A smart offer is an offer that is bound to a smart contract. This is in contrast with on-the-fly offers that are bound directly to an EOA (i.e., an Externally-Owned Account, cf. ethereum.org -> Account Types).
The contract bound to the smart offer is referred to as the maker contract. It implements an offer logic, which is called during trade settlement to source liquidity for the trade and to perform other maker-specific logic.