Yes, typically with a personal auto insurance policy if you loan your car out to a friend from time to time your car insurance coverages would extend to them while they operate your vehicle. Policies though do differ so before allowing a friend or other driver that is not listed on your policy to drive your car you should read through the terms of your policy and speak to your agent to clarify this issue.
There are exclusions in some policies that allow an insurer to only cover those listed on the policy or drivers of a certain age (i.e. over 21 or 25), etc so that is why it is important to read through your policy and see what exclusions if any you have with your policy. If you are unclear on who can driver your car and be totally covered by your coverages ask your agent who should be able to go through your policy with you.
For example the Texas Department of Insurance notes that some car insurance policies won’t cover other people, including family members, unless they are specifically named in the policy. The TX DOI also states though that in most cases, the vehicle owner´s insurance pays for losses, regardless of who was driving.
This means that if you are in wreck while driving a friend´s car, your friend´s policy will usually pay (primary). If you have an auto policy, your insurance would pay any remaining amount that exceeds your friend´s policy limits (secondary insurance). Likewise, if a friend has an accident while driving your car, your policy will pay first, and your friend´s policy will pay any remaining amount that exceeds your policy limits.
So most all auto insurance Liability policies will cover a licensed driver who drives with your permission and conversely, your liability insurance generally will cover you if you drive a friend's car as secondary insurance. Some car insurance policies will however state specifically that no other person is covered when driving your car or have other exclusions. It is always best to check your policy and even with your agent before allowing a friend to drive your car.