There's a name for this, and I even found a web site that helps you create such things for web sites, but I can't find or remember either. The element is on both sides of "Season 2014" on this web site:
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If you ask someone in the publishing world what they are called they will point you to what's called a "Chapter Ornament" or a "Book Ornament". If you want to get further technical on the design process, book designers will refer to them if they are at the beginning of a chapter as a "Chapter Heading Ornament" or at the end of the chapter as a "End of Chapter Ornament".
Some fonts do have ornaments within them and if you reference "BOOK ORNAMENT" by David Moratto he has a good write up on it:
DO BOOK ORNAMENTS PLAY AN IMPORTANT PART IN BOOK DESIGN?
Book ornaments can be described as a book designer tool to further connect a book’s body composition together and enhance the reading experience for the reader. An ornament shouldn’t distract from the content, but rather be an influential silent partner in the background and when noticed should be pleasant to look at. Ornaments can also provide a characteristic feel to a book. Ornaments are very specific and embellish the book’s meaning. A floral ornament would enhance a gardening book but would be a distraction in a book about icebergs.
At the bottom of the article he does address some examples of fonts that do have a selection of ornaments, my particular favorites:
They are also called "flourishes." One of the definitions of the noun form of "flourish" is "a decoration or embellishment, especially in writing," and one of its synonyms is "ornament".
As far as I know, they are called 'embellishments', or smore specifically 'text embellishments'.
Scroll ornament is the term, I believe.
I've always had a hierarchy system in my head. How it got there, I haven't a clue. Thus: text (alpha/numeric/punctuation), dingbats (at times, bullets), flourishes or ornaments (your concern), and finally illustrations -- which have, in my mind, a hierarchy of their own.
They are called glyphs. They are included with some fonts.