Bounties are a new activity in Genshin Impact that were introduced with the 1.1 update. If you visit the City Reputation NPC in Mondstadt or Liyue, you can select a bounty to complete.
These will send you after particular tough or troublesome enemies that the locals have reported in various areas around the map. Talk to the NPC to track the bounties, then check your map to see where they are. The bounty areas will be marked by large yellow circles.
Teleport close to the circle, then head the rest of the way on foot if you need to. When you enter the area, you will be told that you need to find tracks using your Elemental Sight. When you activate the ability, you will notice blue energy leading in a certain direction. This is actually pointing you toward the track you need to find.
Keep spamming the abilty and follow the blue energy, and it will lead you to the track. It can be just about anything, from an enemy, to an item or resource. When you get there, pick and harvest any resources, kill enemies, and general try to find the exact item it is pointing towards. When you do, it will count as a mark being found and you can move on to the next one. When you catch all three, the blue wisps of energy will lead you to the bounty target, and you can take them out, return to the NPC who gave you the quest, and collect your rewards.