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Liability: if your car hit another car (third party), your insurance will pay. But this is mostly an obligatory requirement (and so we have registration plates).
Third party damage: if somebody will hit your car (e.g. in a parking lot), you are responsible (but obviously not if you find who did it, and you get damage compensation directly, but that is outside your insurance).
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According to Investopedia, third party insurance covers two separate things: bodily injury liability and property damage liability.
The first covers injuries to human beings as a result of an accident, including both medical expenses, and things like lost income from being in an accident and unable to work. Such claims can be very large.
Property damage liability covers damage to other people's property, e.g. if you crash into another car or a fixed item like a fence or mailbox - the claim is typically less, but you're probably/hopefully more likely to scrape someone's car than to seriously injure someone.
But you should always read the small print, and if necessary speak to a lawyer or independent advisor.