In the series, Browncoats were named after the simple, brown dusters they wore as uniforms. 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.

The original fans first organized to try to save the series from being canceled by Fox Broadcasting Network. Their efforts included raising money for an ad in Variety magazine and a postcard writing campaign to UPN.

These fans are also sometimes known as "flans". This originated from a slip of the tongue by actor Nathan Fillion during an interview, when talking about Firefly fans.

Browncoats have been known to make Fan Films, and typically wear Firefly t-shirts, and normal the rest of their clothes, and may or may not listen to music pertaining to the series.

Info from Wikipedia (except the last para), image made via a dress up game and paint.