A first-party insurance claim is made by the policyholder for a loss covered by their insurance policy. For instance, you would make a first-party claim if you sustained car damage and injuries in a crash and file a claim with your car insurance company.
On the other hand, a third-party insurance claim is made against another’s insurance policy for a loss the other party caused. For example, if another driver caused your crash, and you make a claim against their insurer, you are filing a third-party claim.