A furbelow has nothing to do with fur. It may be a gathered strip or pleated border to an article of (usually) female clothing, or some showy ornament or trimming.
The word came into English in the early eighteenth century from the French word falbala for a flounce, decoration or trimming on a woman’s petticoat or dress. Though similar words occur in other European languages — such as the German falbel or Spanish farfala — nobody seems to know where it comes from, though I have seen it suggested that it might originate in the Latin faluppa for a valueless thing.
Almost from its first appearance in English, it has had the sense of something ostentatious or showy, to be avoided by ladies of demure disposition:
The costume is simple and plain, — close-fitting upper garments, without fuss of furbelow, and plain close skirts, met at the ankles by high buttoned boots.
Lippincott's Magazine, Dec. 1885
These days it rarely turns up at all, but when it does it usually forms part of the set phrase frills and furbelows, which doesn’t by any means always refer to clothing:
We — of course — ate our way through a full, delicious, well-judged three courses and thought there were just enough frills and furbelows to reveal the kitchen's talent without being so tricksy that it would put off repeat visits.
’ (London), 19 Nov. 2011.