A dinkus is a line of characters commonly used to mark a break in prose. Its standard manifestation is a line of three asterisks centered on the page.
Usually, scene breaks are denoted by an extra space between paragraphs. This is simple, unobtrusive, and still the predominant method in works today.
The dinkus can be useful where a scene break would be ambiguous. For example, in printed work, if the extra spacing between scenes flows over a page, dinkuses are sometimes used in lieu of an extra space to make the scene break more obvious.
Electronic books, by contrast, live in a dimension free from the tyranny of the printed page, and will either have no dinkuses at all, or much more rarely, one at each scene break.
While scene breaks are an obvious application of the dinkus, dinkuses have no specific rules about where they should be used. They are simply a typographical element that can be employed anywhere that a visual breaking up of text might be desired.
For example, if you are writing something non-fiction that could be logically broken into sections smaller than chapters, a dinkus could be used to signal the start and end of each segment to your reader. It can also be used to get out of writing transitions.
Dinkuses come in many forms. The old-school form is the asterism (⁂), which is a pyramidal take on the now-flattened dinkus. But a dinkus could just as well be a line, fleuron (❦), or even a small illustration.
Use dinkuses sparingly. If you have few in-chapter scene transitions, consider employing a dinkus to help those transitions stand out, and make it clear to your reader that the time or place has changed within the same chapter. If you frequently flit between different scenes and perspectives, however, fall back on the old method of an extra space between paragraphs. Your readers will catch onto the flow.
But above all, the most common use of dinkuses in the modern era is to either imbue a work with an old-timey vibe, or to serve as trivia for being something that sounds a bit rude.