Alain Connes and Mathilde Marcolli have posted a new survey paper on Arxiv, A walk in the noncommutative garden. There are many contenders for the title of noncommutative geometry, but Connes’ flavor is the most successful. Most of Connes’ examples are variants of the same basic idea: when a group acts nicely on a space, you can define a new space by collapsing each orbit of the group action to a single point (this construction is known as the quotient space of the action). Unfortunately, most group actions are not nice.
Connes and Marcolli describe an alternative construction. By a theorem of Gelfand, you can study spaces by instead studying its ring of continuous functions (see this Wikipedia article for precise details). Gelfand’s result puts the commutative in commutative geometry. For group actions that have badly behaved quotients, Connes introduced a noncommutative ring that functions as the analogue of the quotient space.