Insurance is a type of contract. The policyholder, or "insured," pays a premium to the insurance company (the "insurer"). In exchange, the company promises to pay for certain types of losses ("coverage") if they occur.
Insurance coverage and payouts depend on the type of insurance and the nature of the claim. Auto insurance and homeowner's insurance cover property damage. If you're in an auto accident, collision insurance will pay for repairs but not medical bills. You need a different type of coverage for that.
In personal injury and property damage cases, attorneys work on a contingency basis so that attorney's fees are part of the award if you win your case. A contingency fee agreement may also include court costs to reimburse filing fees.