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Windows Azure is the foundation for the Azure Services Platform. At its core, Windows Azure is Windows Server 2008 and Hyper-V with additions to make it very scalable. Microsoft mentioned that some of the advances made with Windows Azure will come back into the commercial version of Windows.
One obvious example is the "Live Migration" feature announced in Windows 2008 R2 Hyper-V. This allows a system administrator to move a running Hyper-V Virtual Machine from one physical server to another without interruption. The need for this functionality was most likely driven by the needs for Microsoft to enable high availability in their cloud data centers while still being able to bring machines on and offline at will.
The Diagram above (from David Chappell's Whitepaper) shows detail of the Windows Azure landscape. As you can see the Windows Azure Fabric is the central abstraction here. Think of it as the HAL for the Cloud. The Fabric is responsible for making a huge cluster of physical machines operate as one large node.
Sitting on top of the Fabric are the Compute and Storage Containers (for lack of a better term). The Compute Container is where your application code will reside, both Web Role and Worker Role (the platform is extensible and will allow other roles to be defined