The bgs is the best part of the game imo.
These three sites, especially the last one, really helped me to learn the bgs.
Influence is a way for factions to have or gain control over a star system. Many factions can share influence within a system. The first row in the system's faction lists the controlling faction for a system and always controls the controlling station, which is usually the biggest in the system...
elite-dangerous.fandom.com
The Background Simulation is dynamic, complex and reflects the evolution of the galaxy with ever changing power and system influence of factions in response to player behaviour. Every Station, Surface Port and Settlement has its own demands and supply that change dynamically. The BGS does not...
elite-dangerous.fandom.com
novaforce.com
Any 7 day event, like wars or elections, will be won once one faction has won 4 days.
So if you win 4 days in a row, the conflict will end after the 4th day.
If the other faction has won 3 in a row, then you need to win the next 4 to win overall. That conflict will have lasted 7 days.
Basically the first faction to win any 4 days out of the 7, wins the conflict, whether that takes 4, 5, 6 or 7 days.