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A franchise agreement is a contract between a person who wants to own a franchise of some chain of businesses and the entity that owns the chain. As an example, this would be an agreement between the McDonald's corporation and a person who wants to own a local McDonald's store.
In general, such an agreement will include the franchisee (the person wanting the franchise) the right to use the trademarks of the overall corporation. In return, the franchisee will pay some sort of a fee and will agree to a number of conditions that are meant to ensure that each franchise maintains the standards expected by the parent corporation.