Your policy only covers ur vehicle & the the drivers of said vehicle. Other specified drivers can be named on ur policy (like a spouse, parent, room mate etc..). The insurance does not carry over to other vehicles unless u include another vehicle on ur policy (second car, child is car). You will pay for coverage on each vehicle that u have designated to the policy.
If u r looking to rent a car for travel purposes, the lease will include a charge for for the insurance coverage of the vehicle that you're renting. Companies like Hertz, have deals with insurance companies to pay a flat rate for their rental cars. The amount of coverage that u will pay depends on the amount of travel or time of lease. Again, this surcharge will be included in ur rental fee.
If u drive someone else' car & have an accident, the insurance of the owner or policy holder of that vehicle will have to cover the damage. Again be careful, because while a few states offer policies in which any driver of a vehicle is coverered, most states require u to list the names of drivers on ur policy. If a listed person has a lot of violations or is a high-risk driver, having them on ur policy will cost u more.
* It is true that in states that do not require auto insurance, u r free to opt out of the insurance offered by the leasing company; however, if anything happens to the vehicle, u r financially liable. If you're spending the money to lease a vehicle, then the extra $10-20 daily coverage is certainly worth the expense & will save u the hassle of taking out year long rental coverage with ur own insurance co. After all, how many times throughtout the year do u find yourself renting a vehicle?