Cohomology is a powerful mathematical technology for classifying differential forms. In the 1960s, work by Sir Michael Atiyah, Isadore Singer, Alexandre Grothendieck, and Friedrich Hirzebruch generalized coholomogy from differential forms to vector bundles, a subject that is now known as K-theory.
    Witten has argued that K-theory is relevant to string theory for classifying D-brane charges. D-brane objects in string theory carry a type of charge called Ramond-Ramond charge. Ramond-Ramond fields are differential forms, and their charges should be classifed by ordinary cohomology. But gauge fields propagate on D-branes, and gauge fields give rise to vector bundles. This suggests that D-brane charge classification requires a generalization of cohomology to vector bundles -- hence K-theory.

Overview of K-theory Applied to Strings by Edward Witten

D-branes and K-theory by Edward Witten

    Geometry was originally developed to describe physical space that we can see and measure. After modern mathematics was freed from Euclid's Fifth Axiom by Gauss and Bolyai, Riemann added to modern geometry the abstract notion of a manifold M with points that are labeled by local coordinates that are real numbers, with some metric tensor that determines an extremal length between two points on the manifold.
    Much of the progress in 20th century physics was in applying this modern notion of geometry to spacetime, or to quantum gauge field theory.
    In the quest to develop a notion of quantum geometry, as far back as 1947, people were trying to quantize spacetime so that the coordinates would not be ordinary real numbers, but somehow elevated to quantum operators obeying some nontrivial quantum commutation relations. Hence the term "noncommutative geometry," or NCG for short.
    The current interest in NCG among physicists of the 21st century has been stimulated by work by French mathematician Alain Connes.

Two Lectures on D-Geometry and Noncommutative Geometry by Michael R. Douglas

Noncommutative Geometry and Matrix Theory: Compactification on Tori by Alain Connes, Michael R. Douglas, Albert Schwarz

String Theory and Noncommutative Geometry by Edward Witten and Nathan Seiberg.

Non-commutative spaces in physics and mathematics by Daniela Bigatti

Noncommutative Geometry for Pedestrians by J.Madore