There's no real definition for an FF.net muse. It can be anything from a favorite canon character, to a real-life significant other, to a co-author or friend who gave them a really good story idea via IM (I'm partial to this one). Sometimes, it's even used to refer to the story idea itself.

What would be really impressive is someone identifying exactly which Greek muse is inspiring them in their current fic. :D

Fanwriters (or, sometimes, fanartists) will often refer to their favored fictional characters as "muses" - some of them even have conversations with their muses in author's notes (a practice generally reviled by most of those who don't).

Fandom RPers will often refer to any of the characters they RP as muses, as well.

It is fairly common for a male artist to consider their girlfriend a muse - I'm not sure how often the reverse is true. This is especially true for painters/photographers/musicians.

But as stated, a muse can be anything. It can be a dog, a close friend. A ball you like bouncing.

In short, the Greek Muses were minor goddesses/spirits who inspired art and music and poetry. The term has come to be used by writers as meaning the source of their creative ability; if your Muse is feeling lazy, nothing gets done. Some people play on it and add that tf she's indulged with extra viewings of her(/your) favourite show or book, she can be bribed into giving more inspiration. Chocolate also works. Some people say they have multiple Muses - sometimes one for each fandom, or one for writing and one for painting, or whatever.

Long version of the original Greek stories are here.

Hope that helps. :)