If you are in an accident and have suffered an injury or your property has damaged, then you can file an insurance claim (as long as you have insurance) in order to receive financial compensation from your insurer. Depending on whether or not you live in a "liability jurisdiction," your insurance may have to financially cover another person's injuries or damages if you were liable for them.
Different states have different rules in regards to this. Some states are no-fault jurisdictions, which means that your own insurance pays for your own injuries or damages, even if they were caused by someone else - and vice versa. However, in a few states, insurers can offer you the option of choosing one of the other when you first purchase your policy, which is what personal insurance companies in New Jersey are able to do.
If you've chosen a traditional liability system, then you may dispute liability if there is a disagreement over who was at fault for an accident. This can happen if two parties don't know who was at fault (for example, in a car accident). Each party's insurance company will investigate the accident in order to determine which party is liable. The insurance company of the party found to be liable will then have to pay all the costs or damages and injuries that were involved for both parties.