To capture territory in New World, you will have to declare war! A Company can only declare war on a territory that conflicts. A territory in conflict is an area where an opposing faction has performed several PvP Faction Missions, raising their own Influence to 100% and reducing the Influence of the faction currently in control of that territory.
If all those conditions are fulfilled you will be able to declare war with a press of a button. You will need to buy a territory first which costs around 10,000 gold.
Related: New World Leatherworking Leveling Guide.
Remember to prepare for battle and upgrade your fort. A company will win the battle by either capturing or defending the fort at the designated territory.
When Rally Point is captured, it cannot be re-captured in the same battle by the defending side. They serve the purpose of respawn points of the attacking side. Both sides will have access to siege weapons to break down the fort of the defending side.
For more articles and guides on New World make sure to check out New World Harvesting Leveling Guide.