Browncoats were soldiers who fought for the Independents, who lost to the Alliance in the Unification War. They were named after the simple, brown leather trench coats they wore as uniforms. They also wore red scarves around their necks. The simplicity of their gear was greatly due to the fact that technology and resources were scarce on the Outer Planets. Conversely, the Alliance's soldiers were called "Purple Bellies", because of the purple tinted body armor they wore on their chests.
After the war's end, clothiers made good money dyeing brown coats blue or gray as folks wanted to forget the past and let the past forget them. Those that still 'wear the brown' do it on purpose.