Configuring Vanity URLs for use with Windows Azure Websites
The form of the default name for an Azure Website is along the lines of:
But you'd very probably like something like:
To get that with a Windows Azure website - you need to configure DNS and make entries in to your DNS registrar's zone file for the domain you want to use. You need to use some special "Azure Websites Verify" (awverify) CNAME records to prove to Azure that you actually own the domain.
This video shows you how to do it: