Liability means that you are covered if you cause an accident or damage to property and you are sued for those damages. Liability is the bare minimum insurance and is required in all states.
Full coverage means that you not only have liability insurance, but also are covered if you get in an accident, even if you caused it. You would still have to pay a deductible (you can chose to set yours at $250, $500, and $1000 usually), but your insurance company will cover all other damages. Full coverage also includes comprehensive coverage which applies to things like theft and damage from natural causes (tree falling on car, hitting a deer, etc.).
Insurance varies state to state, so speak to a representative and he or she can give you more details that are specific to you.
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