General character related / writing goal questions - This is primarily useful if you're not sure if you should keep a character / plot thread / idea. By focusing on what you want, you can more clearly argue that something fits your goals or that you may need to change things.
1. Are you aiming for exploration (how do these people tick? what's the world like?), narrative (this is the story of a lovely lady,) thematic (pride leads to -), or something else? If you don't have a set plot, what do you think you'd like to do with this person / place? What'd be the most fun for you to write?
2. How does this person / place _presently_ help this goal? What do you think would make them fit this goal better?
Charting out desires or goals - These are tricky. Obviously, some stories will not have these. These can be a short term goal, or a long term goal. The important thing is that they help you think about what someone wants.
When _____ ends, _____ will have happened.
No matter what happens, _____ will never change.
_____ will change.
_____ may change.
For people - You'll notice that I emphasize change and conflict a lot. This is mostly because a story tends to be easier to write if there's a driving force behind the plot or something that interests you in the dynamics. If you think of your characters as chess pieces, then consider these conflicts / changes as sort of opening gambits. A well done one may make the story almost write itself. Even if you're not writing a long story with a lot of conflict, little ones can help define the character's strengths and weaknesses.
1) Would it be more interesting if -
2) If X changed, what'd happen?
3) What are their stable rocks in their life? Who do they turn to?
4) What do they want? What _should_ they want?
5) What're they really good at? If you were trying to send them on a job / mission / whatever, what'd you think they'd do well? How could they fail at this?
6) What would they be horrible at? How could they succeed?
7) Who would they like / trust? What're the characteristics of this person (can be hypothetical)? Who wouldn't they like / trust? Why? How could this opinion change (or is it unlikely)?
8) Pick a skill this person has. How did they learn it? Tell me about the teacher. When did they learn it? What was their life like then? Why did they learn it?
9) Tactical planning. Toss this character in a fight (verbal, physical, political, e-mail flamewar, pick your poison.) What'd happen? Why? Who could be involved that'd change this answer?
10) If I was their enemy, what could I easily find out? What would be hard to find out? What would they try to hide? What's their greatest vunerability?
For places - A lot of these would work better in longer stories since you'd obviously have more reason to play in an area. One thing I like to focus on is the micro / macro level of details in places. You can have something important to a small area (the crazy neighbor) and something important to the entire town (the river's rising). A mix of these can help make a complex place that is brimming with things that you can use for your plots.
1) Tell me one legend / famous event / recent occurance in the news.
2) Liesure - what do people do? Music? Religion? Where are there private spaces? Public spaces? Is there a night life?
3) Pick a character. Give me their description of the place? What do they know? What don't they know?
4) Why are your characters here? What's important about this place? What makes it interesting plotwise?
5) Settings - list some common haunts for your characters. Tell me about the history of the places, if interesting or relevant.
6) If I'm someone new in the area, what will I see? If I'm a local, what _don't_ I notice?
7) Let's chart out this place on an axis. Big or little? Spread out or compact? etc.
8) Character X needs to obtain an item in this town. What will they see on the route? Is it an easy trip? How would they plan this trip?
9) Basic needs. Food, rest, drink, work.
10) Plot hooks. On a personal scale for your characters or residents, what might be some small secrets / wants / goals that could come up? If your characters met someone, what could they discover? On the neighborhood / family / small community level, what kinds of things could be discovered? This could include things like landmarks, bits of history, or allies / enemies. On the city-wide / regional / universal level, what could be discovered in this place that could apply elsewhere?