Soviets are often portrayed in a furry, almost-cylindrical hat. My friend and I call it the “furry Russian hat,” but putting a name on it would be nice. Can anyone help?
It’s not the proper name, but from Douglas Adams’ Meaning of Liff:
GLINSK (n.) A hat which politicans buy to go to Russia in.
One style is an ushanka. Is that the style you had in mind?
A shapka.
Ushanka literally means a hat with ear flaps. A shapka is the puffy fur hat commonly associated with Russia.
In the movie “Gorky Park”, they were called sable hats, since that is the fur used to make them.
I call it The Costanza.
Many shapkas, when brought back to the US, are used to hide the spare roll of toilet paper. 
Ha!
Well, I have a rabbit fur shapka with ushankas. Of course, now that I live in Texas I don’t really get the chance to wear it, but when I lived in Minneapolis, I loved it. I’ve also got a neat Red Army thing that looks cool on the front of it.
A shapka is literally any type of winter hat, including wool beanies. At least in Moscow.
Err, I suppose I should actually be useful.
An “ushanka” is a fur/faux fur hat with ear flaps that can be tied up on the sides of the hat or tied down to cover the ears.
An “ushanka” is a type of “mehovaya shlyapa” or “mehovaya shapka” which means “fur hat”. Typically “mehovaya shapka” since “shlyapa”, I believe, was originally referred to gentlemen’s tophats and became generic word for “hat” due to recent misuse.
And is “shapka” related to or derived from the French word “chapeau”?
According to Vasmer’s Etymological Dictionary the first mention of the word is in 1327, which is pretty old. The suggested etymology is from “sсhарёl” from some form of old german (the abbreviation escapes me), “chapel” from old french or “cappa” from latin. Possible routes are direct or through polish. So, while precise etymology is unknown, it’s definitely related to the romance word.
Well according to this site it derives from French chape, meaning roof. The -ka suffix is a diminutive, so you’re literally wearing a “rooflet”.
On preview, I see groman has a cite that gives a different etymology, which sounds more authoritative.
In any case, the shapka-ushanka is declining in popularity these days. You won’t see all that many males older than 8 or younger than 60 (other than soldiers, policemen, and bomzhi [bums/homeless]) wearing one in Moscow these days.