Reputation represents how the general public perceives you, whether positively or negatively. This perception precedes you, speaking on your behalf when you are absent and determining how you can expect to be treated by those who have heard of you. Reputation means different things to different types of characters, reflected in the social and cultural values of Korvosa and its different regions. A character who embodies the qualities of a hero in one region may be perceived as villainous or disreputable in another.
The one thing to keep in mind when it comes to reputation and fame is: It doesn't grow If nobody knows about your character's heroic deeds. Whether your character defeats a mighty foe, crafts a powerful magic item or makes a significant discovery, it either needs to happen in public or you have to ensure word gets out.
Three factors determine your reputation: Fame, Sphere of Influence, and Prestige Points. Your Fame increases and decreases depending on your actions. Your current Fame determines your overall reputation and maximum potential for cashing in on your fame (for a heroic character) or infamy (for a villainous character). Sphere of Influence defines the places where you can apply the benefits of your reputation. You can reap the benefits of your reputation by spending Prestige Points on awards, including temporary bonuses and favors. It can get you out of sticky situations, attract an imp as a servant for a while, selling loot at a higher price or even get a wizards to teach her some cantrips for a day and much more.
I will add specific rewards your characterers can access with their prestige once they have build some reputation.