One of the ways in which we can get bewitched by language is by assuming that if we correctly use a word in a range of contexts, there must be some common essence shared by the things we refer to. Wittgenstein uses the example of 'game'. One approach to defining 'game' would be to look for all the non-trivial features that are shared by the things we call games. But to assume that there must be some underlying essence of all games, is, Wittgenstein tells us, a mistake. Instead of the single uniting feature, what we find if we actually 'look and see' is a complex pattern of overlapping resemblances between games. He coined the term 'family resemblance' to refer to this pattern: this a reference to the visual resemblances between genetically related members of a family. Often such relations don't share a single distinctive feature, but rather one is like another in hair colour, others have similar eye colour or cheekbones, etc. So that as with games there is simply this complex pattern of overlapping visual resemblances (albeit with an underlying genetic cause). The key passages where he discusses this idea are in Philosophical Investigations (1.66 and 1.67).
This approach is anti-essentialist in that Wittgenstein rejects the idea that all concepts appropriately used refer to a common underlying essence that make that thing what it is. That view was, for example, explicit in Plato's notion of the Forms.
A number of philosophers, notably Maurice Weitz, applied the notion of a family resemblance term to the question 'What is Art?' Weitz diagnosed the difficulty that previous philosophers had had with defining art as arising out of a misplaced quest for the essence of art. An example of a writer who assumed that art must have an essence is Clive Bell, the Bloomsbury critic, who in his book Art (1914) declared that '...either all works of visual art have some common quality, or when we speak of 'works of art' we gibber' (p.7-8). A Wittgensteian would point out that this a false dichotomy: that there is a third option, namely that art is a family resemblance term.
For a discussion of neo-Wittgensteinian theories of art, see Nigel Warburton The Art Question (Routledge, 2003) Chapter 3.
These links are useful for an overview of Wittgenstein's life and philosophy:
Numerous further links at episteme.com