The Map
Press the touchpad (PS4) or Back button (Xbox) to enter the map menu. If you want to clear up the fog of a portion of the map, visit a high viewpoint (marked by eagles) and synchronize at its position. When you do so, you'll also be able to Fast Travel to its position, a very handy perk. There are various icons on the map. Story icons are large golden indicators, while Conquest Activities - missions you need to do in order to take control of a portion of London - are marked by emblems. In each borough, four Conquest Activities must be finished. This unlocks a gang fight (and a boss fight afterwards); complete this and the borough will be yours.
Conquest Activities
Child Liberation: Requires you to free sets of children working in a factory.
Templar Hunt: Requires you to assassinate Templar(s).
Gang Stronghold: Requires you to clear out a Blighter stronghold.
Bounty Hunt: Requires you to kidnap an enemy and place them into a carriage, then driving them to the designated waypoint.
These activities, as mentioned, unlock a Gang War. This is a fight with you and your Rooks against the Blighters, followed by a Boss Fight.