Running a Business Discussion Boards

Let's say that I would like to sell an application that uses some open source code inside.

The open source definition says that "the program must include source code". http://opensource.org/docs/osd[^]

However, the open source FAQ says that if the source code isn't distributed, then the license does not even apply to that situation. http://opensource.org/faq#non-distribution[^]

Except, in some circumstances, the source code has to be made available. What circumstances? I'm confused...

The FAQ is quite clear: If you don't distribute source code, then what you are distributing cannot meaningfully be called "Open Source". So you cannot claim it to be open source if you do not make all the source code freely available.

One of these days I'm going to think of a really clever signature.

It is not necessary to publish the source code. But when someone who bought your solution asks for the source code, you have to provide it to him.
Better contact the authors of the code you use and get a different license from them, or find source code which is published under a different license.