AZURE is microsoft cloud computing platform. Essentially you sign up and provision a VM based on their baselines and your custom configurations and pay on time usage.
You can store data and build and connect applications on their hosted infrastructure.
Think of it as rent a server. I have a couple in the CLOUD :).
http://www.microsoft.com/windowsazure/
Great video explantion by Microsoft - http://www.microsoft.com/windowsazure/videoPlayer/videoPlayer.aspx?w=640&h=480&vid=WinAzPlatform_720x480_FINAL_101609.wmv
Cheers,
Hades666