Dashicons are what’s known as an icon font and were added to the core of WordPress with the release of 3.8. The icons are vector based so they can be as large or small as you want without losing quality. Plugin authors can use CSS, HTML, or a Glyph for use within Photoshop to display an icon. While 3.8 had 167 icons, WordPress 3.9 shipped with 30 new Dashicons bringing the total to 197.
The icons cover Media, TinyMCE, WordPress.org, Sorting, Widgets, Alerts, and Miscellaneous. Some plugin authors have already opted out of using a bitmap image and are using a Dashicon to represent their plugin within the WordPress admin menu. If none of the Dashicons match your use case, try Genericons instead. Genereicons is also an icon font but has icons that are not focused on WordPress.
Are there any “average WP user understandable” instructions on how to use them? Did try to find out but didn’t understand a single line of code and how to implement those icons in posts. Maybe it’s only me but not so sure.