When applying for insurance what does two-party mean?

You may have misconstrued the Words. With respect to property and casualty insurance, there are the concepts of first-party coverage and third-party coverage. First-party coverage is that which you obtain to protect your pwn property. An example of this would be the collision coverage on your car, which pays to repair the car irrespective of whether you or another person damages it. Conversely, an example of third-party coverage woukd be liability coverage. This pays for the damages of a third party (property damage or bodily injury) if you are found legally liable for the damages. All of that said, if what you really mean is that someone used the term "two-party" to you in applying for insurance, my best assessment would be that you and another person are insured under the same policy. Since you posped this question in the health insurance category, it may mean that you and a spouse are insured under the same policy, which would not be uncommon. If you can provide further facts, I'd be pleased to further clarify.