Claims link First Ones with settlements they do not currently control. These claims indicate which settlements a First One has or believes to have a right of ownership to, but does not currently control.
Claims are always shown on both the character and the settlement details pages.
There are two kinds of claims. Simple claims can be made at any time on any settlement and require only a herald to announce it. These claims are primarily for political purposes and have no game-mechanical effects.
Enforceable claims however, are of a different kind. The most common way these come into existence is when a settlement is taken away from a First One by force, through the take control action. De jure, the First One is still the rightful owner of the settlement, and will have peasants, administrators and soldiers inside who are loyal to him. This makes it much easier for him to re-take control, speeding the process up dramatically.
The same system is used when an estate is granted in fief. The granter retains an enforceable claim, allowing him to take the estate back from his vassal should that one not fulfill his obligations.
Claims can be withdrawn by the claim holder at any time.
Enforceable claims are a major liability and estate holders are well-advised to take care of them. Unfortunately for them, there are only two ways for a claim to disappear: Either the claim holder withdraws it, or they die.
This gives a huge incentive for the winning side in a war to offer surrender terms to the losers, usually including the giving up of some of their enforceable claims in exchange for peace (and thus keeping their remaining estates). Of course, since claims once given up are gone, there is some negotiation required here.
These claims also make it dangerous to be an aggressive conqueror with no regard for chivalry. Having several destroyed enemies with nothing to lose can make maintaining your empire next to impossible. Having them surrender or join you as vassals is often much less risky.