Having a claim to a title means you've got a justification for it being yours. Claims come in two varieties, weak and strong. Claims are used as Casus Belli for war. The difference between a strong and a weak claim can be determined by the graphics, and mousing over a claim will indicate whether it will currently be inherited.
 Strong Claims
Strong claims are, as the name implies, the most useful type of claim. A strong claim can be pushed against anyone by almost anyone. The exception is that you cannot push a vassal or courtier woman's claim upon a title if the title uses Agnatic succession law (If you are female, you may however press your personal claims no matter what the succession law is). Only the second and third in line to a title will get strong claims upon the titleholder's death. (There is one exception: under gavelkind succession, the primary heir does not get claims on titles that are passed to his siblings if they become his vassals, although he does if they become independent.)
Strong claims can also be gained via the Fabricate Claim councillor mission as well as the Forge Claim plots. A character who loses a title (through war or intrigue) will also get a strong claim on their former territory.
Children of strong claim holders inherit weak claims. The one exception are claims granted by the councillor mission, which are not inherited unless pressed in war.
 Weak Claims
Weak claims are not as useful as strong claims, but still have some use. A weak claim can only be pushed if a title is either held by a woman, is in regency, a succession crisis is ongoing, or they're 2nd or 3rd in line to the title. Like with strong claims, you cannot put a woman on a throne if the succession is Agnatic. All children (unless 2nd or 3rd in line) of a titleholder get weak claims upon the titleholder's death, and their children can inherit the claim, and they can be inherited from holders of strong claims. The claim has to be pushed if the children's children are to inherit the claim.
 Pressing a claim
A claim is "pressed" when war is declared using that claim as a casus belli. Pressing a claim "renews" it, ensuring that it will be inherited by the claimholder's children. Even if the war ends in white peace or inconclusively, the claim remains pressed. It is sometimes useful to press a claim of a courtier and then get white peace, so that that courtier's children - belonging to your dynasty, perhaps - will inherit the claim.
 Impact of claims on AI
Your AI vassals with claims on one of your titles will have a lower opinion of you. Additionally, the AI is much more reluctant to release a prisoner with a claim on their title.