Persistence is a game quality that preserves progress of players actions. Types.

1.World-level persistence: all characters, NPCs and actions alter world features and events. Single player RPG with saved maps/world state are typical expression of this level.
MMOs are typical "persistent world" games, without major player-driven changes to the (mostly static) world.

2.Character-level persistence: Only character is preserved, as world is instanced(diablo series) or regenerated from scratch each game(e.g. procedurally).Characters equip/items/stats are always preserved between games. Most RPGs fall within within this category.

3.Stat-persistence: The game keeps track of scores/levels or some other character stats while the game runs(progression), but each new game doesn't preserve earlier status/items/achievements won in game(except ratings).
The key difference between char-persistent and stat-persistent games is "does playing new game preserve earlier(old game) gained advantage(that is char-persistence)?"
Typical for E-sports games where player start anew and develops a character: such as FPS/MOBA(e.g. League of Legends).
Games which preserve skills/items for in-game advantage fall into #2.

4.Zero persistence: the player has to replay each game as new, better stats/items/achievements cannot be gained. Typical for puzzle games, strategy and board games(though they might save non-ingame stats such ELO ratings ).
Pure skill games(e.g. Chess/Go) always fall in this category, as they require all players to have a fair start.
Pure skill RPGs are also possible, but without any progression: all stats/items/skills have to be distributed at start and game would not reward any action.