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 insurance is a coverage that helps pay to repair or replace your car if it's damaged in an accident with another vehicle or object, such as a fence or a tree. If you're leasing or financing your car, collision coverage is typically required by the lender. If your car is paid off, collision is an optional coverage on your car insurance policy.