This was a commercial name given to a certain unique phenotype that has become more or less "official" due to its widespread use. The "Armageddon" betta is typically a bicolor with a blue/black/turquoise body and yellow or orange fins. But this is where it gets interesting. The Armageddon is unique because of its ability to express red or orange at the same time as yellow (non red), making for a dramatic and colorful effect. It is the hope of those involved in Project Armageddon to get a better grasp of how this works on a genetic level, how it was originally created, and how to best perpetuate it.
According to PG Saga, an Armageddon betta can be achieved by crossing a blue/yellow or purple/yellow bicolor betta to a copper/red. Richard Kent Laughery had very similar fish randomly appear in his Purple line. In my own experience, the effect can be seen as early as the F1 by crossing orange dalmatian to blue/yellow bicolor.
Here are some examples from the web of Armageddon bettas.