A first party claim is made against your own insurance company for benefits provided by your policy. A third party claim is made against someone else. For example, if someone causes damage to your property and you make a claim against that individual, that is a third party claim.
The distinction between a first party and third party claim is significant. In a first party claim your rights are determined by the contract. In a third party claim your rights are determined by the laws of the state in which you live.