Making a claim with “the other driver’s” insurance company? That’s a third-party claim. Driving a company car and sustain an injury? You’re making a third-party claim there, too. Injured while driving your own car in the course of your job? A third-party claim with your employer’s insurance company would be your mechanism for recovery of medical and repair costs. Any claim made to an insurance company other than your own is considered a third-party claim.
Third-party claims are much more prevalent in “fault” states than in “no-fault” states. While no-fault generally require an injured party to first recover from their own insurance company, fault states have no such requirement. In fault states, if you sustain an injury to your person or damage to your vehicle in a car accident you can make a claim with “the other driver’s” insurance company -- provided they were at fault for the accident.
Comprehensive insurance is a coverage that helps pay to replace or repair your vehicle if it's stolen or damaged in an incident that's not a collision. Comprehensive typically covers damage from fire, vandalism or falling objects (like a tree or hail). If you're financing or leasing your car, your lender likely requires comprehensive coverage. If you own your vehicle outright, it's an optional coverage on your car insurance policy.