A business-facing test is a test that's intended to be used as an aid to communicating with the non-programming members of a development team such as customers, users, business analysts and the like. When automated, they describe the system in domain-oriented terms, ignoring the component architecture of the system itself. Business-facing tests are often used as acceptance criteria, having such tests pass indicates the system provides the functionality that the customer expects.
Automated business-facing tests are often represented in some form of DomainSpecificLanguage, since this helps communication with non-programmers and also helps give programmers a mechanism that helps them step back from the details of the code. Tools like Cucumber and Twist help design such DSLs and provide mechanisms to bind them to the system under test.
Business-facing tests are commonly implemented as BroadStackTests since their user-oriented expression suggests treating the system under test as a black box. However there are significant advantages to implementing business-facing tests as ComponentTests since this often results in easier maintenance and faster execution.
I'm a big fan of automated testing, but it's important to recognize that manual tests play a significant role in business-facing testing. Techniques such as exploratory testing and usability testing are inherently manual activities and are essential parts of a well-balanced test portfolio.