A spindle is a narrow wooden rod that's used when spinning wool. If your aunt has a spinning wheel for making yarn from the wool of her sheep, it must have a spindle on it for twisting the fibers.
The noun spindle can be used to talk about a number of different forms of wooden rod or dowel, particularly if they turn or spin. A yarn spinner's spindle is one kind — another is a chair leg or a banister rail that was made from being turned on a lathe. In fact, spindle and spin share a common root word, the Old English spinnan, "draw out and twist fibers into a thread."