An insurance policy is based on a legally binding agreement between the insurance company and that of the policy holder. In the terms of the agreement it is decided what is being insured against (health, automobile, property), and the specific terms of the policy. For example in the case of auto insurance, the cover may only kick in once $500 or more damage is done. In the case of property insurance, the property may only be insured if a specific event occurs like a fire. When the predetermined event occurs, once the minimum deductible is reached, or if there is an option to cash out early (mainly with life insurance), the policy holder or their representing lawyer will contact the insurance company requesting payment. This process is known as an insurance claim.
In most cases an official insurance statement of claim document must be completed before formal proceedings can go ahead. In some cases the poor filling out of statement has resulted in the claim not going ahead due to legal technicalities. This however will only happen if the insurance company feels cheated in some way by the claim. The majority will happily fulfill a claim whether they are car insurance claims, life insurance claims, accident insurance claim - it is after all their business reputation at stake.
It is called a claim because at the initial conversation, it is the policy holder’s claim that they reach the terms for a payout, it is only after further investigation and proof that the claim actually results in a payout. Sometimes insurance companies will dispute the claim if it clearly doesn’t meet the terms laid out in the contract or if they have reason to believe the policy holder is committing fraud, by either faking a claim or deliberately causing a payout event. An example of this might be deliberately setting a business on fire to reap the insurance money.