An agreement to cover a loss resulting from the insured’s liability to a third party, such as a loss incurred by a driver who injures a pedestrian.  The insured’s claim under the policy arises once the insured’s liability to a third party has been asserted.

In Plain English:

A 3rd party insurance claim is one that is pursued against the insurance company of a person or business which has harmed you. Examples would be: a personal injury claim for damages you make against the insurance company of a car that hit you in the crosswalk or a personal injury claim against a drug company that manufactured a medicine which caused you physical problems.