It's clear from some posts that not everybody has worked with the Backgound Sim before. If you haven't, don't worry as even those who have are constantly fighting it as there are many many Broken Bits that have been around for a Long Time and are still Not Fixed.
Some of what follows will be known by all, hopefully, some will be new. There is also no guidebook and a great deal of controversy about what really happens, so some of this may be wrong or at least contentious.
There's a TL:DR at the end for the impatient.
Here is A Beginner's Guide to the BGS
Each system has from one to six factions (although higher numbers have been spotted). There are one or two leading factions: the Top Dog and the Chief Influence (a title you won't see anywhere else). In Azrael they are currently one and the same. The Top Dog owns the best station and controls the system, but the Top Dog doesn't always have the most influence. For a number of reasons, influence can be distributed to the point where the Top Dog has just 1%, yet still be in control, while the Chief Influence has the most power.
There are six factions in Azrael, but Mobius members are asked to support only one, the current Top Dog: The Order of Mobius, Cooperative, Independent at 80.5% influence (tonight's figures).
The other factions are in competition:
Az. Empire Consulate, Patronage, Empire, 8.2%
Az. Patron's Principles, Patronage, Empire, 7.2%
Az. Crimson Galactic Ind, Corporate, Independent, 2.2%
Az. Mafia, Anarchy, Independent, 1%
Labour of Az., Democracy, Independent, 1%.
By definition, there are only 100 points of influence, although the sharp-witted will notice that here we have 100.1 thanks to rounding issues. You can find influence levels in the right-hand status panel of your HUD (where figures are rounded down) and in the System Map (where the figures are roundly fudged).
A faction gets to be Chief Influence by taking influence from the other factions.
• Every time you successfully conclude a mission for a faction, that faction takes a proportion of influence - small, medium or large - from all the other factions (and you gain a small, medium or large amount of kudos with that faction depending on the mission).
• If you take a mission and abandon it, the commissioning faction loses influence and the Chief Influence gains it.
• Every time you kill a pirate ship in a RES the pirate's faction loses influence to the Chief Influence (and you lose a little kudos from them and gain a little with the Chief Influence).
There are two ways of getting to be Top Dog:
• If two factions are on par with equal influence levels they could go to Civil War (if they are of opposite type(Corporate and Anarchy, for example), or Election if the are of the same type (Patronage and Patronage).
• A faction gaining 70% of the influence without previously challenging for the system can also trigger a Civil War or Election.
As you know, you get a side to win a Civil War by picking a side in a CZ and killing ships of the opposition. When a ship is destroyed the killer faction takes some influence from the opposition but nobody else gains or loses. To win an election you take missions for your the faction you support. The loser of a Civil War or Election hands over their most valuable station to the winner (if they have one).
You can cash in your Bounty vouchers, Combat bonds and Exploration data at any station in the system, but make sure you choose the right one: the owner of the station takes a little influence from the other factions each time you make a transaction of this type.
As influence can be modified through actions in Solo, Open and Group, there are often changes in the daily tick (when all influences are updated) that are difficult to explain. Sometimes a Wing (or worse) will descend on a system and just take the profitable missions. Some missions are rumoured to have an extraordinary effect on influence and cause really wild swings in the fortunes of all factions - 60% overnight is not unknown, so the Chief Influence can change several times over a week.
Azrael is in a difficult position as it's being exploited by an Empire Power, which means that pledged players will be constantly popping in and out as they play their power games. This is just to remind you how insignificant you are as there is nothing you can do to defend aginst their depradations that doesn't work against you.
TL:DR - Don't do ANYTHING for any faction other that the one you wish to support: The Order of Mobius, of course.