Xbox Linux is a project that ported the Linux operating system to the Xbox video game console. Because the Xbox uses a digital signature system to prevent the public from running unsigned code, one must either use a modchip, or a softmod. Originally, modchips were the only option; however, it was later demonstrated that the TSOP chip on which the Xbox's BIOS is held may be reflashed. This way, one may flash on the "cromwell" BIOS, which was developed legally by the Xbox Linux project. Catalyzed by a large cash prize for the first team to provide the possibility of booting Linux on an Xbox without the need of a hardware hack, numerous software-only hacks were also found. For example, a buffer overflow was found in the game 007: Agent Under Fire that allowed the booting of a Linux loader ("xbeboot") straight from a save game.
The Xbox is essentially a PC with a custom 733 MHz Intel Mobile Celeron processor, a 10 GB hard drive (8 GB of which is accessible to the user), 64MB of RAM (although on all earlier boxes this is upgradable to 128MB), and 4 USB ports. (The controller ports are actually USB 1.1 ports with a modified connector.) These specifications are enough to run several readily available Linux distributions.
From the Xbox-Linux home page:
The Xbox is a legacy-free PC by Microsoft that consists of an Intel Celeron 733 MHz CPU, an nVidia GeForce 3MX, 64 MB of RAM, a 8/10 GB hard disk, a DVD drive and 10/100 Ethernet. As on every PC, you can run Linux on it.
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 or 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.
Read more about Xbox Linux: Uses, Softmod, List of Distributions