your risk of being sued by someone: in legal terms, some third "party" in addition to you (first party) and your insurer (second party). When someone talks about a third-party claim against you, that means you are being sued by a person who holds you responsible for injuring them or damaging their property. The person suing can be your own passenger in a crash you caused. The damage you cause to another person - the third party - is covered under Third Party Liability insurance.