The Compass of Order and Chaos is an artifact that Wilhelm can look into and see the order in phenomena happening before it occurs. It is the artifact found in the very beginning of Episode I, a glyph with a small silver colored Zohar that was put in the middle by T. Masuda during early 20XX AD.
In the game a mysterious girl named Nephilim quotes that Phenomena occurs in waves created by little ripples in time, or peoples actions, that are always changing. Obviously Wilhelm can see past this and tell what happens no matter what.
In Episode I it is referred to only as the Compass of Order, because only the Order ring on the Compass is presently working. The Chaos ring of the Compass has not spun in some time.
In the epilogue of Episode I, the Compass of Order goes haywire, glowing brightly and wildly changing directions when chaos's hands begin to glow, possibly suggesting that it can't predict chaos's intervention. However, Wilhelm seems hardly put off by this; instead, he appears fascinated.
In Episode III it is revealed that the Compass is a terminal interface for Zarathustra, which resembles the Compass on a colossal scale. It also collects and processes the data from Program Canaan and sends it to Zarathustra, enabling it to track the progress of those under surveillance by Canaan units. In the Official Perfect Guidebook, it is stated further that the Compass is actually part of Wilhelm, it is his manifested ability to predict all possible futures. He has used it since the beginning of time to help him guide humanity from behind the scenes.
Because it is a relic of God, it was not meant to be touched by human hands. When Voyager is tricked by Canaan into accessing the "source" of Wilhelm's power, he connects to the Compass and by extension Zarathustra and is overwhelmed and obliterated by its power, along with Canaan.
The Compass of Order and Chaos also serves as the logo for the title of Episode III.