According to the Microsoft Azure Services Platform site, "Windows Azure is a cloud services operating system that serves as the development, service hosting and service management environment for the Azure Services Platform." This definition, while concise, requires some imagination.
Drawing out the definition a bit more, Windows Azure is an application development and runtime environment (.NET in this case), hosted and managed by Microsoft (within Microsoft "owned and operated" data centers), that enables IT organizations and software vendors to offer software as a service (SaaS). To put it more succinctly, Windows Azure is an application platform offered as a service (PaaS).
PaaS, such as Azure, is at least, one part of cloud computing (albeit nascent). Looking at the market today, several PaaS choices are available including: Google's App Engine, Salesforce.com's Force.com, and 's Cloud Computing strategy.