Windows Azure provides an emulator environment that runs on your development machine. You may either run a Windows Azure app in that emulator, or publish to Windows Azure. There's no option to deploy a Windows Azure application to Windows Server.
While some of the services offered are similar (such as cache), Windows Server AppFabric is not equivalent to the services provided by Windows Azure, and there's no equivalent fabric controller that runs on Windows Server.
EDIT: Adding clarification per @Asmaa's comments.
Building a private cloud is not the same as creating a local version of Windows Azure. The Private Cloud you reference is essentially a set of Windows Servers running virtual machines managed by Hyper-V, and some type of local storage. You'd be fully responsible for replication, disaster recovery, durable storage (where storage survives even with a disk crash), scaling, etc.
Windows Azure provides the abovementioned features, but only within its data centers. As @Avkash mentioned, there is a Windows Azure Appliance, but it there are only a handful in existence and they're huge (as in made for a data center).
If the objective is to run your code either locally or in Windows Azure, this can be done, but not by pushing the Windows Azure package to your Windows Servers. You'd need to build separate solutions to run your code in the two environments. You can reuse your core code if it's packaged correctly, but the notion of the Role Environment, Diagnostics, and other Windows Azure features would need to be replaced with something else.
One last thing: if you want to build a hybrid app, where some of it is deployed locally and some to Windows Azure, then there are certainly ways to bridge the two. See this article for more info.
