What happens if a person has liability insurance and totals another car worth more than their policy can cover?

The Insurance co will pay the policy limit. You can sue the driver for the rest, obtain a judgment and enforce in order to collect all your damages.

Word of advise here, the Insurance co will ask you to sign a release of all claims, meaning that once you receive the check you can not sue the driver, it is standard practice for them.

Do not sign a release unless paid in full, you're not obligated to sign it in order to collect all your damages, either.