An Xbox with Linux can be: a full desktop computer with mouse and keyboard, a web/email box connected to TV, a server or router, a node in a cluster.
You can either dual-boot or use Linux only; in the latter case, you can replace both IDE devices. And yes, you can connect the Xbox to a VGA monitor.
The name “Xbox” is short for the development name of the product: “DirectX Box,” named for Microsoft’s graphics API. Since the Xbox uses DirectX heavily, it’s apparently not hard to emulate an Xbox on a PC.