The Pontryagin classes are characteristic classes on the classifying space of the orthogonal group and, by pullback, on the base of any bundle with structural group the orthogonal group. The latter is where they were originally defined.
B O(n) \to B U(n) \,.
The twisted differential c-structures corresponding to Pontryagin class include
twisted differential string structure for the first fractional Pontryagin class ;
twisted differential fivebrane structure for the second fractional Pontryagin class .
Classical textbook references are
Paul Bressler, The first Pontryagin class, math.AT/0509563
Ivan Panin, Charles Walter, Quaternionic Grassmannians and Pontryagin classes in algebraic geometry, arxiv/1011.0649
A brief introduction is in chapter 23, section 7