Claim
Worlds can be claimed for an empire. A claim is staked by raising a flag of your empire. To do this, hold a flag in a ready inventory slot and select it as the current item in hand. Look at the exact spot to put the flag pole and click with the mouse.
When a world is claimed, all cities and buildings on that world become part of the claiming empire, if they were not already.
Construction
Once a world is claimed, only the empire holding the claim can construct buildings, cities and bases on the world.
A world does not have to be claimed to build a city. An unclaimed world can hosts cities of multiple empires.
A world must be claimed to construct a military headquarters.
Quit Claim
A world can be unclaimed. This is called quitting the claim. The Governance window (default F12), Cities and Bases tab, provides a button to quit the claim of the world you are currently on.
When a claim is quit, cities on the world revert to their former empires, if they were part of a different empire when the claim was staked.
Stealing Claims
You can steal the claim to worlds from other empires simply by claiming the world yourself. The claim to a world with a military base cannot be stolen. Claims of friend empires cannot be stolen.
Claims of neutral empires can be stolen with confirmation. The first attempt to steal the claim from a neutral empire warns that the world is already claimed. A second attempt to claim that world within one minute steals the claim.