From Emil Chronicle Wiki

Prestige (often called Reputation or Fame) is the measurement of how well a character is known and liked by NPCs.

Editor Note: This article is largely speculative. We have no explanation of the prestige mechanic from any official sources.

[edit] Usage of prestige

A certain amount of prestige is required as a prerequisite for some quests, such as the catty quests. Prestige is also used to expand Rings.

[edit] Gaining prestige

Prestige is earned through completing quests. It is believed that prestige gain is based on the color difficulty rating of quest. Yellow quests reward 2 prestige points, white quests reward 1, and gray quests do not reward any. (Speculative! More information needed)

[edit] Losing prestige

You can lose prestige for failing a quest. (More information needed)