Reputation is a gameplay mechanic in Octopath Traveler.
Reputation is an invisible statistic that affects how townspeople interact with you in each town. Each town tracks reputation separately and upon entering a town for the first time, you have a neutral reputation. Reputation can be increased or decreased through various means.
Reputation is decreased when you fail to perform a rogue Path Action. These path actions are Scrutinize, Allure, Steal and Provoke. If your reputation drops five times, you will be unable to interact with the townspeople in that town or perform Path Actions until you increase your reputation.
Reputation can be increased in two ways. Firstly, when you complete a quest for an NPC, your reputation in their town will increase. This appears to include main quests as well as side stories. Otherwise, you can increase your reputation by paying the owner town's tavern to spread good rumours about you. This is the only way to increase your reputation if you already have a bad reputation in that town (i.e. you cannot interact with townsfolk). Currently, there is no known benefit for having a high reputation.