Personal liability insurance provides coverage if someone should sue you, as an individual. It pays for legal (lawyer) costs, as well as paying out (up to the policy limits) if you should lose. Usually the policies state what is covered or not covered, e.g., car accidents or accidents on your property. Standard polices from your Homeowner's insurance company usually exclude work (A&P, CFI) or flying.
If you have insurance on your aircraft part of the premium is for personal liability ("Liability").
Unfortunately our peers (juries) tend to hand out large awards in aviation accidents. IMHO sufficiently high limits are just not available for any price, leave alone for a reasonable price. I'm a CFI and will not sign off students anymore for solo; too much uninsured risk.
One of the posts suggested forming a corporation (with almost no assets) so that if the corporation was sued it wouldn't lose much. Whether or not this shields the individual from a law suit depends on the state. Here in CA this approach is useless for an individual. If you have a partnership and it's your partner that screwed up, an LLC may provide you some protection.