What does an eagle represent on a coat of arms?
An eagle represents freedom. In its claws it holds olive branches and arrows with its head turned towards the olive branches so as to promote peace before war. It used to be turned towards the arrows but was later changed so America was not viewed as imperialistic.
Why was the bald eagle chosen as the national bird of the US and the golden eagle chosen for the Mexican coat of arms?
It's in fact a black and red eagle on golden background, similar to the national colors. The eagle is the symbol of the Roman Empire. The First Reich, the so called Holly Roman Empire of German Nation, had the same coat of arms. That's the reason why both German countries, Germany and Austria, have a black eagle as coat of arms ( the Austrian version looks different of course ).
It has evolved according to the government in turn; with the exception of the coat of arms, the colors and proportions have remained unchanged for the last 200 years. The coat of arms has evolved from an imperial symbol akin to the Spanish coat of arms, to a Roman look-alike eagle, to the present coat of arms, last adopted in 1968.