Quests from Peacekeepers can be obtained after completing Becoming a Peacekeeper at the Bounty Board. They are generally procedurally generated, the exception being the quests to upgrade the Peacekeeper Station.
Peacekeeping quests usually involve flying between different planets and/or even systems. Repair your FTL engine, accumulate fuel and prepare to fly a lot.
Kill a monster
A rather straightforward quest where you are directed to a planet where a certain monster is terrorizing local population.
Eliminate a local gang
The Peacekeepers determined that a small criminal organization is threatening local population. Neutralize it.
Completing this quest requires you to walk around the target planet first so that your scanner can pinpoint the location of the bandit base. Note that it can easily be hidden underground (or on the water, in case of ocean planets). The base is protected (in a sense that its blocks are under effects similar to a working Shield Generator) until you defeat the last enemy standing.
Apprehend a local gang leader
The most prolonged (and, consequently, most rewarding) quest usually found at the top of the Bounty Board. You will need to travel between different bandit bases (starting at the one the Peacekeepers were able to pinpoint), defeating bandits and searching for clues as to where their leader hides.
Clues can take form of:
- Notes found inside the base
- Information stored in computers
- Hand prints found and scanned via Matter Manipulator (search for green prints)
- Disks containing a recording of a gang lackey accidentally saying the current location of the gang leader.
When enough clues have been found, the Peacekeepers are able to deduce the next location. Often bandits also share this information if you are... persuasive enough.
Finally, you need to defeat the leader of this syndicate. Some gang leaders are bold enough to offer a bribe for you to let them go. Consider it at your own risk (note that closing the bribe window via [Esc] is qualified as declining a bribe).