My favorite Cold Weather (winter) hat looks something like this .
I was just searching for an example of my hat, and saw that style called "trapper", "trooper", or "bomber" on different web sites.
I had always called it my "Russian hat", because you always see some guy on TV named Leonid, standing on a podium watching a parade of tanks go by.

I "customize" mine a little bit, by putting on a regular "ball cap" first, and then putting the Russian Hat on over that. That way, I have a "Russian hat WITH BRIM" !!!