I thought it was, "La plume de ma tante," you know, from the song.
Hmm.. I have been waiting 1336 days for this phrase to come up in Duolingo... but when it does, it's different!
To clarify: un stylo is the general word for a writing-instrument which uses ink, ie 'a pen'. Because they are oldest types and in some ways resemble a quill pen, a pen you dip in an ink-pot and a fountain pen = 'un stylo plume'. A ball-point pen = 'un stylo-bille'. A felt/fibre-tip pen = 'un stylo-feutre'.
They were the primary sort of pen before ballpoints came out, where there would be a nib, similar in a lot of ways to even older quill pens, attached to a cartridge which contained liquid ink, the ink would be drawn into the nib and could be written or drawn with. They are still used today and I have at least one. They are especially used for calligraphy, but need care as they tend to blot a lot more (hence the 'blotting pad' you may have encountered in older literature). It is a successor to the dipping pens where you had to dip the pen into ink continually, instead the pens had a 'fountain' of ink contained within the pen.
You're making it sound all so exotic! Fountain pens are my go-to for every-day writing, no special occasion required. I like them because it is something permanent, with only the ink that needs replacing, and that ink comes in a fantastic array of colours.
And although they can get pricey, an inexpensive entry-level Cross will run you about $15.-- at Staples, or a better-quality one will go for just under $50.-- (in most cases, you get what you pay for).
Worth checking out.