In some cases, if a family member is visiting and has permission from the insured to drive the family vehicle, there will be coverage if there is an accident, but the coverage may be limited. All policies should be reviewed to determine if there are any excluded drivers and any limitations on coverage for anyone driving the car that is not specifically named on the policy.
The answer to whether insurance follows the car or driver depends on many variables, most notably the kind of insurance coverage being referred to. There are coverages that follow the car and coverages that follow the driver. In general, auto insurance follows the car instead of the driver, but the specifics of a claim can differ since insurance laws and coverage vary depending on the policy, coverage and state being dealt with.