Build a Shipyard or a Military Base in a sector to claim it for your faction. There may not be more than one Shipyard or Military Base in a sector. Building a control station in a sector under the control of another faction is frowned upon, and the other faction may try to destroy it. They may even declare war on you.
If there is no Military Base or Shipyard in the sector, the faction with the most Factories controls the sector.
If you are officially at war with another faction, you can claim a sector that's neighboring yours by selecting it in the Empire Manager (F1) / War Management panel. There are some prerequisites:
Pirate sectors may not be claimed and will forever be Pirate sectors. However, Factories and Stations can still be built in Pirate sectors.