In ”standard” geometry it is possible to associate to every geometric space an algebraic structure, called an algebra; a set with a well-defined addition and multiplication. From this algebra it is then possible to reconstruct the geometric space, and often it is possible to describe most geometric concepts in an algebraic way instead.
One important property of this algebra is that it is commutative, meaning that you can switch places in the multiplication: ab=ba. In noncommutative geometry, we start with an algebra, which we not necessarily assume to be commutative, and study the geometric concepts via the algebraic point of view.
