Microsoft Azure is the collective name for Microsoft's cloud computing services that provide IaaS and PaaS service models. In terms of deployment models, Azure services would be classed as public; however, it's possible to install Azure Pack (http://www.microsoft.com/en-us/server-cloud/products/windows-azure-pack/) in a private data center that offers a private cloud model.

Microsoft Azure IaaS comprises of a number of globally distributed data centers that host virtualized servers controlled by the Azure Fabric Controller. When we host systems on Azure, we become tenants and pay for our share of processing and network resources that we use through the subscription we choose. In this layer, we can make use of services such as virtual machines, disk storage, and network services.

Microsoft Azure PaaS services are the main entry point for most developers, where we are offered a set of tools and services that allow us to develop and deploy scalable and robust systems such as websites, worker roles, and mobile services.

Microsoft Azure (formerly known as Windows Azure) was first announced in 2008 and was available as Community Technical Preview (CTP); then, it became commercially available from 2010. Since then, the number of services and features has continually grown to where we are now.

At the time of this writing, Windows Azure has been renamed Microsoft Azure as part of a rebranding exercise to move the services away from being tightly associated with Windows server operating systems, databases, and platforms, as Azure can support operating systems such as Linux that run on virtual machines, Oracle databases, Node.js, and PHP websites, to name a few. You may see the term Windows Azure still being used in documentation and resources for quite a long time. It's the same product, just with a different name, so it's likely to be still valid.