Can I use the same service bus Azure uses and host it in my own environment? If not what service buses are simple and effective and work well with low latency/bandwidth?
Microsoft also has Service Bus for Windows Server http://msdn.microsoft.com/en-us/library/windowsazure/jj193022(v=azure.10).aspx.
Clearly the architecture of Azure is slightly different then the average Windows Server installation; so as to whether they're the "same", I don't know.
In terms of the API, they seem to have parity (identical) as far as I've been able to tell. http://msdn.microsoft.com/en-us/library/windowsazure/jj542433(v=azure.10).aspx I have had a problem trying to get a client running on Windows 8, so I haven't taken it much further than a simple test.
Depending on what you need, alternatives might be NServiceBus, MassTransit, message queues like RabbitMQ, ZeroMQ, etc.