Reputation
In Elite Dangerous players can change their Reputation with the three major factions to either Allied, Friendly, Neutral, Unfriendly, or Hostile. All players start out as Neutral with all three major factions and by doing missions or other influence increasing activities can improve their reputation with that faction. When a player performs activities that would decrease a faction's influence (e.g. pirating or failing missions) the player's reputation with that faction will decrease. When a player has a negative (current value unknown) reputation with a faction they will be attacked on sight by NPCs that represent that faction.
Local Factions and Reputation
Who's in control of a system has a major bearing on how your actions affect your reputation with them and the other minor factions. There are usually several minor factions vying for control in each system, but outside the immediate local space around their starports and outposts, only one can claim to be "in control" of a system. This is displayed in the system view when you've no stellar body selected.
Normally it's the faction with the highest influence in the system, except in special situations like civil war.
This is important for criminal law, because each faction only cares about crimes committed in its own jurisdiction. It doesn't matter if you are an infamous pirate according to say, the "Bangwa Freedom Party", if you are in the jurisdiction of the "Bagwa Independents" and you've committed no crimes against them, you are an innocent pilot.
In addition, most ships are aligned with a minor faction (you can see which one by performing a basic scan).
So if a ship is wanted in the current jurisdiction, but is aligned with a different minor faction for which it has no criminal record, then destroying it will cause you to lose reputation with the faction it is aligned with; in their opinion, you've just destroyed an innocent citizen of theirs.
Notes
- Reputation is different than Rank