Reputation shows how popular the inn is among different Social Groups.
Every guest belongs to one of the five Social Groups. If you meet the preferences of their Group, they will leave satisfied. This in turn increases your Reputation with this Group (see the panel on the left side of the screen), as well as influence in the town from which the guest hails, and your general level of fame.
You can increase the satisfaction of a given type of guests by managing the menu, style, and services of your inn according to their liking. The needs of your guests depend on their Social Group and your fame. You can even kick out guests of a given social group, if you wish to focus on another/others.