When you express your right to something, you're making a claim to it, like telling your sister, "This bowl of cereal is mine."
A claim is when you express your right to something that belongs to you, like your medical records or the deed to your home. When you make a claim or claim something, you're demanding it or saying it’s true. People claim dependents and deductions on their taxes. In court, you could claim you deserve money from an employer who cheated you. You could claim you can juggle chainsaws. With any kind of claim, you're going to have to prove it.