It is said that in theBrotherhood of Steel symbol, the gears represent their engineering knowledge, the sword is their will to defend themselves, the wings are the uplifting hope, and the circle is the wholeness that makes it a brotherhood.

While the original Brotherhood uses different colors for different elements of the symbol (blue for the wings, gray for the sword and black for the gears), the Midwestern Brotherhood of Steel Detachment uses several monocolor variations. A blue version is used by the Knights and Paladins, a black one by the Scribes and an orange one by the Elders. The golden/yellow variant seems to symbolize the Brotherhood as a whole. The Midwestern version is also reversed horizontally - the large gear is to the right of the smaller ones, not to the left.

Yes, I'm quite aware I've already made this post 3 times over before. Its called a "re-cap".
Enjoy