There is really no such thing as 'full coverage". No insurance company should advertise full coverage. Agents may use it as a term to reference physical damage coverage along with the state required coverage.
Most people use this term to refer to physical damage coverage. Physical damage coverage consists of collision and comprehensive coverage. Generally, collision insurance covers damage to your automobile caused by collision with another object or by upset. Comprehensive insurance covers damage to your automobile from vandalism, theft or glass breakage. Most lienholders require physical damage coverage if you are financing or leasing your vehicle.

You may be required to carry a specific amount of liability insurance if you are leasing the vehicle rather than financing it. Liability coverage consists of bodily injury coverage and property damage coverage. Bodily Injury Liability covers other people's bodily injuries or death for which you are responsible. Property Damage Liability covers you if your car damages someone else's property.

You can learn more about coverages and definitions here. And you can shop for an insurance policy by getting a car insurance quote here.

Comment Update: Thank you for the added comments to improve our answers. One of our carriers, The Hartford has great extended benefits for "full coverage". Here are some of their added benefits:

• Auto & Home Account Policy Savings
• 24 x 7 Claim Reporting Service
• Countrywide Claim Service
• 1st Hartford Not-At-Fault Accident Forgiveness
• 24-Hour Roadside Assistance
• Legal Representation for Accidents
• No Deductible Windshield Repair Service
• Home Ownership Credit
• Full Range of Deductible Options
• Full Range of Coverage Limits
• Customer Repair Service Program (CRSP) w/Life of Vehicle Guarantee
• Paid in Full or Monthly EFT Required *
• Disappearing Collision Deductible *
• Full Value Replacement Cost Coverage *
• $100 Collision Deductible Waiver w/CRSP *
• $10,000 Accidental Death Benefit *
• Deductible Waiver for Not-At-Fault Accident *
• Accident Forgiveness *
• Renewal Assurance *
• Emergency Expense Coverage *

* May vary by program selection
May Vary By State

Comment Update: Regardless of a policy with physical damage or not, it is subject to the guidelines of the policy regarding payment of claims.