With general liability insurance, the insurer is required to defend you and pay for the damage caused by you in a covered liability claim. These claims include bodily injury, property damage, personal injury, and advertising injury (damage from slander or false advertising). The insurance company will also cover compensatory and general damages. Punitive damages aren't covered because they're considered punishment for intentional acts.
General liability policies state a maximum amount the insurer will pay for that policy period - say $1 million. If your business is successfully sued for $1.5 million, the insurer will pay the agreed $1 million and you would be responsible for paying $500,000. To cover this situation most companies purchase umbrella insurance, which picks up where general liability coverage ends.