Your policy only covers your vehicle and the the drivers of said vehicle. Other specified drivers can be named on your policy (like a spouse, parent, room mate etc..). The insurance does not carry over to other vehicles unless you include another vehicle on your policy (second car, child's car). You will pay for coverage on each vehicle that you have designated to the policy.
If you are 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 you will pay depends on the amount of travel or time of lease. Again, this surcharge will be included in your rental fee.
If you drive someone else' car and 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 you to list the names of drivers on your policy. If a listed person has a lot of violations or is a high-risk driver, having them on your policy will cost you more.
* It is true that in states that do not require auto insurance, you are free to opt out of the insurance offered by the leasing company; however, if anything happens to the vehicle, you are financially liable. If you're spending the money to lease a vehicle, then the extra $10-20 daily coverage is certainly worth the expense and will save you the hassle of taking out year long rental coverage with your own insurance co. After all, how many times throughtout the year do you find yourself renting a vehicle?