A life or death struggle seems to be the most common occurrence, though I suppose if we're talking about mental or emotional stress, the return of a suppressed memory or being forced to relive a painful event in one's life would suffice as well.

As for what Personas are for, they're to help us defeat the demons and other creatures that stalk the streets of this city at night. When the clock strikes midnight, there's a time called the 13th Hour when those sorts of creatures emerge.