Business cases are documents that argue for a specific action, and provide substantiations, rationales, justifications, and other support in order to compel the sponsor into taking a desired action.
Business cases are mostly drafted for commercial purposes where you have an amount getting invested, and the business goes into justifying the investment and coming up with the return on investment. For example, if you are going to invest X amount of money into a project, sub project, or a new business, how much time will it take for you to recover the money and what is the rate of profit you would start seeing thereafter?
In IT, we write business cases all the time, depending on the domain you are in. Some commercial business cases include ...