I know this, but I can't recollect at all.
What is this thing called?
They are usually centered in a book or writing and I'm certain they have a name but I can't remember for the life of me. They also come in all shapes, colors and sizes.
Flourish, Filigree, or Ornament are common terms.
Usage doesn't generally change the name. Simply because it's been used by the book designer to separate content, there's no special term I'm aware of for that specific type of usage.
There are some fleurons available in the Unicode specification under the Dingbats block (PDF).
Many of the other Wingdings fleurons were recently added to the Unicode specification under the Ornamental Dingbats (PDF) block.
Screenshot of text because most of those glyphs are not yet supported (in my browser anyway)