Carbon Fiber has a very high strength to weight ratio, which has provided a huge advantage when used in the construction of modern airplanes. Having several advantages over steel or aluminum has allowed for much higher use of these carbon fiber structures in airplanes.
The following is a great video sharing more information and visual representation of carbon fiber applications, structures, and manufacturing of the material itself.
This video discusses the Airbus airplanes, and Boeing is using this technology as well.
The lower weight of carbon fiber composites provides the advantage of lower fuel usage and associated fuel emissions.