The purpose of difficulty naming is to inform players how difficult a map is. This article lists common conventions as well as other alternatives that have been gathered throughout the years. However, as long as the set of difficulty names is descriptive and clearly indicates progression of difficulty, you may use any desired kind of scheme.
These difficulty names are the most commonly used for each game mode.
Note: The special difficulty name Marathon historically allowed mappers to submit maps of longer songs, and thus became popular for single-difficulty mapsets. Today, the name has no special meaning and is typically only used for very long songs or song compilations.
Mapsets of songs that originated from other rhythm games often borrow that game's difficulty naming scheme. Usage of these difficulty names is acceptable for any song. Some of the common naming schemes that made their way into osu! are listed here:
Note: Large mapsets of SOUND VOLTEX songs often use difficulty names from multiple versions of the game, even if the song does not appear in all of them.