Xbox:HDMI Output

Modern TV's may not have component/composite inputs and/or you may prefer to connect your Xbox via HDMI.

Fully digital
By tapping directly into the digital signals from the GPU you can skip the on-board encoder. This is very useful on a 1.6 xbox and arguably gives the cleanest picture of any solution on any version of the original xbox.

However, this solution will also require the most soldering and is the of the most expensive solutions.

Plug in & play converters
Another option to use HDMI is to use a 'plug in & play' adapter. The pro's of this is that there are a lot of adapters available both cheaper and even some high end ones, however none will bypass the on-board encoder.

This solution will take no soldering and has a broad range of quality and price.

Comparison
There are a lot of adapters available and the most known/branded ones are added to this list. Other options include "Kaico adapter", "Pound", "Hyperkin", "Portta" or modding a "Wii2HDMI", however these are more for convenience as for signal quality and will be beaten by a decent component cable.

One special note regarding the "Pound" adapter is that for certain revisions of the xbox, it's better to use the externally powered cable as the on board filtering caps of the xbox aren't sufficient and can cause problems.