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Failed binary is an informal term for a star system that developed like a binary star except that one of the two YSOs did not achieve the mass of a star (currently 75 Jupiter masses is a commonly-used threshold), leaving a brown dwarf or gas giant. The term may be used to refer to an entire system, or may be used as an adjective, or may be used specifically for the lower mass object. The phrase failed binary system more explicitly means the entire system.
The term failed binary is used in discussions of the formation history of star systems and the potential signs and consequences of such histories apparent in observation. The term implies primordial binary stars form in a manner differently than a gas giant within a planetary system, but planet formation theories now suggest there may be multiple ways planets to form, and possibly some observed giant planets are failed binaries.
The term is sometimes used for the solar system, to point out Jupiter's similarity to a star, but though discussions admit the similarity, they generally assert the term does not apply because Jupiter's formation is presumed to be different from the formation of binary stars.