One of the most important rules regarding allowing someone else to drive your car is believe it or not- where they go! It can become a nightmare if you don't know what your insurance policy covers.
If the person says "I'm just going to the gas station down the street", and does exactly that and comes straight back but has an accident that is their fault either coming or going, then your insurance company is liable for any claims made by the driver and occupants of the other vehicle. Property damage claims are included.
If the person on the other hand decided to go somewhere else or to another place or on another street that is not enroute to or from the gas station your insurance company can and most likely will refuse to pay. They will assume that the driver was not acting under your direction. In this case the driver of the vehicle will be reponsible for the payment of any claims made. If you are willing to help, you can.The driver is only covered under your policy as long as they are going to or from where they told you they were going before they left in the car.
The other fact to consider is that most insurance policies state in the fine print that additional drivers to whom you give permisssion to drive your car must be at least twenty five years old. So if you decide to send someone to the store even from your own immediate family members in the household, sending a young person could have very serious consequences even if the store is just a couple blocks away.
If your policy is only a simple basic liability one then your vehicle would not be covered in any case. On a policy that is considered to be more than the minimum then your vehicle may be covered. You should ask your agent to explain some of the fine print and make sure you understand it.
Financial responsibility laws differ from state to state so be sure of what your responsibility is regarding other drivers.