Eudemon system is a feature of Chaos. Having High-quality Eudemons is an indication of approved or superior status.

A eudemon is something that you attach to your weapon.  Not just any weapon, you need an awakened weapon.  Once your weapon is greater than lvl 20, you should understand.  Once you add a eudemon to a weapon, you can use the eudemon to add to your stats.

Different quality of eudemon has different appearance. Here is a picture of  High-quality (from quality 13 to quality 46). More features in Chaos is waiting for you to explore.

Following link is an Eudemon Guide:

http://co.enjoymmo.com/2009/04/eudemon-guide/