In ways to push and extend the limits of beatmapping, Expert was invented/created to be beatmaps like Insane but go beyond.
It could be the extra 5th difficulty (if beatmaps are formatted in ) or the 4th (if beatmaps are formatted in ). These maps, like Insane, quote, "generally exemplifications of the mapper's skill, and are the difficulties where creativity can be easily shown."
On the osu!beatmap listing page (for example: Rise Against - The Good Left Undone (pishifat) [Extreme]), one may tell it is an Expert map with this icon:
Characteristically, expert difficulties are difficult maps, usually featuring both streams and jumps. A map receives the expert icon when its star rating reaches more than 5.25 stars.
An issue with Expert maps, like Insane maps, is that they can easily be Overmapped to increase the map's difficulty.
Note: this is only a general suggestion. The difficulty setting will vary depending on the song itself and mapper's own style.
Potential difficulty settings for Expert.