Xbox 360/Xbox Motherboard Replacement
Xboxes have special protection schemes to prevent their users from using them to run software other than Xbox games. Many people hack their Xboxes to overcome this protection and make them run other software, e.g., special editions of Linux. but hacking Xboxes may be illegal in some areas.
However there is another, legal, way to run Linux on an Xbox. Instead of hacking its original, non-standard motherboard, you just replace it with a standard PC motherboard. It then becomes a normal PC, and you can install any operating system designed for normal PCs on it, including any PC version of Linux. The modified Xbox cannot run Xbox games.
The PC motherboard you install into the Xbox should be small in size. There's one such project called i64XBOX (unfortunately, it runs Windows 98) that involves use of a Mini-ITX motherboard , but any other motherboard that is small enough to fit into the Xbox case will also do. Due to size limitations, you probably won't be able to install anything into the motherboard slots, so the motherboard should have integrated video card (and sound card if you want sound, and network card if you want Ethernet, etc).
Warranty, electrical safety and legal issues
This mod will definitely void your Xbox warranty, so it is better to modify a machine with warranty expired. You can cheaply buy a second-hand Xbox with defunct motherboard to perform this mod on it. Dumpster diving may also help if it is legal in your area.
When buying a second-hand Xbox for this mod, make sure that its defunct component is motherboard, not power supply, hard disk drive or anything else. Do not attempt to open the Xbox power supply, unless you have experience in repairing switchmode power supplies, as it has a lot of mains-connected components as well as capacitors which remain being charged even for some time after unplugging.
Please do not attempt to sell your Xbox modified this way, as while performing this mod is probably legal, selling it may still be illegal. See Retropod to get idea of what may happen.
Overcoming the RAM limitations
With traditional way of installing Linux on Xbox, you can face a problem that it has only 64 MB of RAM. You can upgrade it to 128 MB, but this is also too low for most modern Linux distros. With Xbox motherboard replacement, you eliminate this problem, and can install as much RAM as your motherboard supports.