Noncommutative algebraic geometry is a branch of mathematics, and more specifically a direction in noncommutative geometry, that studies the geometric properties of formal duals of non-commutative algebraic objects such as rings as well as geometric objects derived from them (e.g. by gluing along localizations or taking noncommutative stack quotients).
For example, noncommutative algebraic geometry is supposed to extend a notion of an algebraic scheme by suitable gluing of spectra of noncommutative rings; depending on how literally and how generally this aim (and a notion of spectrum) is understood in noncommutative setting, this has been achieved in various level of success. The noncommutative ring generalizes here a commutative ring of regular functions on a commutative scheme. Functions on usual spaces in the traditional (commutative) algebraic geometry have a product defined by pointwise multiplication; as the values of these functions commute, the functions also commute: a times b equals b times a. It is remarkable that viewing noncommutative associative algebras as algebras of functions on "noncommutative" would-be space is a far-reaching geometric intuition, though it formally looks like a fallacy.
Much of the motivation for noncommutative geometry, and in particular for the noncommutative algebraic geometry, is from physics; especially from quantum physics, where the algebras of observables are indeed viewed as noncommutative analogues of functions, hence having the ability to observe their geometric aspects is desirable.
One of the values of the field is that it also provides new techniques to study objects in commutative algebraic geometry such as Brauer groups.
The methods of noncommutative algebraic geometry are analogs of the methods of commutative algebraic geometry, but frequently the foundations are different. Local behavior in commutative algebraic geometry is captured by commutative algebra and especially the study of local rings. These do not have a ring-theoretic analogue in the noncommutative setting; though in a categorical setup one can talk about stacks of local categories of quasicoherent sheaves over noncommutative spectra. Global properties such as those arising from homological algebra and K-theory more frequently carry over to the noncommutative setting.
History
Classical approach: the issue of non-commutative localization
Commutative algebraic geometry begins by constructing the spectrum of a ring. The points of the algebraic variety (or more generally, scheme) are the prime ideals of the ring, and the functions on the algebraic variety are the elements of the ring. A noncommutative ring, however, may not have any proper non-zero two-sided prime ideals. For instance, this is true of the Weyl algebra of polynomial differential operators on affine space: The Weyl algebra is a simple ring. Therefore, one can for instance attempt to replace a prime spectrum by a primitive spectrum: there are also the theory of non-commutative localization as well as descent theory. This works to some extent: for instance, Dixmier's enveloping algebras may be thought of as working out non-commutative algebraic geometry for the primitive spectrum of an enveloping algebra of a Lie algebra. Another work in a similar spirit is Michael Artin’s notes titled “noncommutative rings”, which in part is an attempt to study representation theory from a non-commutative-geometry point of view. The key insight to both approaches is that irreducible representations, or at least primitive ideals, can be thought of as “non-commutative points”.
Modern viewpoint using categories of sheaves
As it turned out, starting from, say, primitive spectra, it was not easy to develop a workable sheaf theory. One might imagine this difficulty is because of a sort of quantum phenomenon: points in a space can influence points far away (and in fact, it is not appropriate to treat points individually and view a space as a mere collection of the points).