Philosophy has a persistent compulsion to look for essences: the single thing that all instances of a concept share and that non-instances lack. Socrates' dialogues are full of this demand — "not just some cases of courage, but what courage is in all cases." Wittgenstein does not deny that some concepts have sharp definitions; he denies that all do, and denies that language requires them. The demand for essence is a philosophical prejudice, not a metaphysical discovery.
Consider board games, card games, ball games, Olympic games. Some are competitive, some not. Some require skill, some luck. Some have winners, some don't. Some are entertaining, some not. Look for the common feature and you will find: there is none. What holds the concept together is not a hidden essential property but a complicated network of similarities and differences — overlapping and criss-crossing like the resemblances among members of a family. A child may have his father's nose, his mother's eyes, his grandfather's build, and his aunt's temperament — yet no feature is shared by all.
Family resemblance concepts are not defective. Vagueness is not a failure of language but a feature appropriate to many of our needs. The concept "game" does not have fixed boundaries because we have not needed to draw them. We could draw them — stipulate that "game" applies to exactly such-and-such — but in doing so we would not be revealing the concept's hidden boundary; we would be legislating a new one. Wittgenstein's point is not that concepts are vague and we must fix them, but that the demand to fix them everywhere misunderstands how language works.
Family resemblance is introduced in §§65–77 of the Investigations. The family-resemblance analysis of games (§66) is the canonical passage. Wittgenstein extends the analysis to "number" (§68) and later to "understanding" and "reading."
