Liability insurance is a very broad superset that covers legal claims against the insured. Many types of insurance include an aspect of liability coverage. For example, a homeowner's insurance policy will normally include liability coverage which protects the insured in the event of a claim brought by someone who slips and falls on the property; automobile insurance also includes an aspect of liability insurance that indemnifies against the harm that a crashing car can cause to others' lives, health, or property. The protection offered by a liability insurance policy is twofold: a legal defense in the event of a lawsuit commenced against the policyholder and indemnification (payment on behalf of the insured) with respect to a settlement or court verdict. Liability policies typically cover only the negligence of the insured, and will not apply to results of wilful or intentional acts by the insured.
Collision coverage has a deductible, which is the amount you pay before your coverage helps pay for your claim. You can typically choose the amount of your deductible when you buy coverage. So, if you choose a $1,000 deductible and your car is later damaged in a covered accident, you'd have to pay $1,000 toward repair costs. Your collision coverage would help pay the rest, up to your coverage limit.