A letter of claim (sometimes known as a letter before action) is a letter asserting wrongdoing of some kind by the recipient. It is most commonly used as the first step in the legal process of a personal injury claim. For example if you slipped on an unmarked wet floor in a restaurant, you or your lawyer may write a letter briefly outlining the facts as you see them, noting that you will be officially perusing a claim against them.
Letters of claim are also often used to notify someone of incomplete or unsatisfactory work on a specific project, or more formally a breach of contract. For example, if you hired someone to decorate your office, but they never finished the job or did not complete it to your specifications, you would write them a letter of claim stating you grievances and possible reparations. Like an injury claim this is the first step before proper legal action is taken.
Letters of claim may also be written for alleged copyright infringement and many other scenarios.