City Reputation is a system that rewards all of your exploration's progression. Whenever you explore around the map and open chests, guide seelies, find elemental oculi, unlock waypoints, etc, you will level up that region's city level. Increasing the reputation level will give you plenty of rewards that are useful for gameplay in general.
How to Unlock
You will get the quest to unlock the City Reputation System once you reach Adventure Rank 25. You will also need to complete the archon quest "The Outlander Who Caught the Wind" Prologue: Act I to unlock the one in Mondstadt. For Liyue you need to complete the archon quest "Farewell, Archaic Lord" Chapter I: Act II.
The quest to unlock the one in Mondstadt is called "Knight of the Realm". Talk to the Knights of Favonius Coordinator NPC, Hertha, near Angel's Share tavern, to unlock the feature.
The quest to unlock the one in Liyue is called "Ministry Missions". Talk to the NPC Ms. Yu, that's behind the blacksmith in Liyue.
How to Increase Reputation
There are 4 different ways to increase reputation: Region quests, World Exploration, Bounties, and Requests. The important part here is that Bounties and Requests are limited to only 3 attempts per week, and the attempts are shared for all cities. If you do 3 in Mondstadt, you will not be able to do 3 again in Liyue.
You can only do up to 3 bounties per week, and the attempts are shared between cities, so doing 3 in Mondstadt means that you won't be able to do it again for Mondstadt and Liyue. The higher the stars, the higher the difficulty and points that you can get. You can only accept 1 bounty mission at a time. Make sure you choose carefully which city to level up first.
Bounty bosses are a little different than going out into the world and just finding that enemy type to kill. Once you accept the bounty, the description will show where they are on the map.
Once you enter the area, you will have a set time limit to find tracks before you can reveal the boss. You can't kill the boss before finding the required tracks. Use your elemental sight, by using middle mouse button on pc, or click on the "eye" icon next to the minimap, to help locate the tracks and find the boss,
Similar to bounties, you can only do up to 3 requests per week, and the attempts are shared between cities. But unlike bounties, there is no difficulty differences between requests. All of them give the same amount of rewards. You are able to take on multiple requests at a time. Once you complete the request, you don't need to return to the board again.
There are plenty of rewards that you can get from the reputation system. The important ones are the blueprints since they are quite useful for exploration. The recipes are unique to the cities, and the attempts are for the bounties and requests are limited, so it's recommended to choose carefully which one you want to focus on first.