A real-world contract, simply stated, is an agreement governing outcomes for actions, given a set of inputs. A contract can range from formal legal contracts (e.g., a financial transaction) to something as simple as the "rules" of a game. Typical actions can be things such as fund transfers (in the case of a financial contract) or game moves (in the case of a game contract).
An EOSIO Smart Contract is software registered on the blockchain and executed on EOSIO nodes, that implements the semantics of a "contract" whose ledger of action requests are being stored on the blockchain. The Smart Contract defines the interface (actions, parameters, data structures) and the code that implements the interface. The code is compiled into a canonical bytecode format that nodes can retrieve and execute. The blockchain stores the transactions (e.g., legal transfers, game moves) of the contract. Each Smart Contract must be accompanied by a Ricardian Contract that defines the legally binding terms and conditions of the contract.