The answer to this question is found at Nvidia's own What is CUDA? article:
CUDA is a parallel computing platform and programming model that makes using a GPU for general purpose computing simple and elegant. The developer still programs in the familiar C, C++, Fortran, or an ever expanding list of supported languages, and incorporates extensions of these languages in the form of a few basic keywords. These keywords let the developer express massive amounts of parallelism and direct the compiler to the portion of the application that maps to the GPU.
People have used GPUs to solve problems such as: