Microsoft Azure is an open and flexible cloud platform that serves as the development, data storing, service hosting and service management environment. Microsoft Azure provides developers with on-demand compute and storage to host, scale, and manage web applications on the internet through Microsoft data-centers.
Cloud computing is a way to deliver computing as a service rather than a product. It is completely based on the Internet. Cloud Computing provides on-demand hardware (like Server), storage resources, services hosting and services management environment and other devices as a utility or resource over a network.