What is a letter of demand to at fault insurance company from attorney?

It is just what it sounds like. However, the letter is addressed to the person who was At Fault in the accident not the insurance company. They may have forwarded a copy of letter to the person's insurance company just so that the adjuster is aware of it. It is part of the process of filing a suit against someone. Attorney's tell their clients this so it seems like they are doing something.