This question has been my hobgoblin in every game I've designed. It's so easy to rely on our experience and punt this question or gloss over it when writing rules, but designing an intentional, thoughtful answer to it improves any game dramatically.
The lazy answer I fall back on is 'when the GM doesn't want to say yes, but he doesn't want to say no either.' We grant authority to the dice to decide what happens. The less experienced a GM is, the more tempting it is to let dice decide everything. But this approach can lead to many undesirable situations.
* Player just gave a long, stirring speech to his PC's followers. Everyone at the table was impressed. Why, oh why did I ask for a Charisma roll? He rolled a 1 of course.
* Player tries an audacious move and everyone at the table is excited. Rules don't give enough guidance for 'what happens when they fail?' so the awesome move just fails and falls flat, disappointing everyone. Not saying success should be guaranteed, I'm saying trying and failing should be consequential and interesting, compared to doing nothing or just taking the path of least resistance.
* Why did I put the letter the PC's absolutely need to move on in this adventure in a locked drawer they failed to open? or hidden so a failed search check de-rails the story? Endless variations of this.
Unless you provide a clear, explicit and well-designed answer to this question in the rules, novice GMs are going to flounder.
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Here are some innovative answers I've seen in other games.
* We roll when there is a clear conflict between a PC and another character. No conflict, no dice.
* Whenever the player wants his PC to do one of the things on this list, we use dice. (Careful design in the list, and what happens on failure.)
* Anytime the players want to, but there's an explicit cost or risk every time you roll, with an explicit upside for success.
* Players spend dice out of a pool.
* Players decide when to roll but must set explicit stakes in the fiction for success or failure.
Many games use the dice mechanic to build tension and propel the game from one conflict to another. Let the consequences for success and failure cascade into the next roll, and the next, with each roll altering the situation in an exciting and meaningful way. You need to lead the GM through this explicitly in the rules.
