It's a history essay, not French dictation.
There are already some good answers to this, especially the accepted answer by Angelo, but it might be useful to add a bit more about the interviewer's perspective.
For context: I work in a technical organisation, and my main job is R&D (doing it myself, managing others doing it), but occasionally I pitch in with recruitment. We have a HR team who help a lot with the logistics, but the actual candidate assessment for our technical stream is mostly done by people who work on that technical side ourselves. As part of that, I've reviewed about a hundred candidate applications and interviewed about forty in person.
When we advertise a position, we will define the skills we're looking for. Usually it will look something like this:
- Technical skills.
- Ability to work well with others.
- Ability to plan and manage work.
- Critical thinking/analytical skills.
- Communication skills.
(For any given position, the requirements will be much more detailed than what I've given above, with details depending on the work area and the level of the position, but that's the gist of it.)
When we interview a candidate, our objective is to assess them on those same criteria. Behavioural questions are a common method for doing that. (There are alternatives; what those are, and whether they're better, is outside the scope of the question.)
So, how do interviewers use those questions?
Rimmer believed there were two kinds of people: the first kind were history essay people, who, started life with a blank sheet, with no score, and accumulated points with every success they achieved. The other kind were the French dictation people: they started off with a hundred per cent, and every mistake they made was deducted from their original perfect score.
-- Grant Naylor, "Infinity Welcomes Careful Drivers".
A couple of the responses to this question have taken the "French dictation" approach, suggesting that these questions are traps used by interviewers looking for excuses to disqualify candidates, and that therefore the best strategy is a grey rock approach that discloses as little as possible.
Maybe there really are some interviewers like that out there, somewhere, but I can't say I've ever met them. For every interview that I've been in (on either side of the table) grey-rocking would have been a huge mistake. My experience has always been much more of a "history essay" model: we want to hire somebody, but we can't afford to hire somebody who doesn't have the required skills, so we need you to convince us.
When my panel interviewed candidates recently, we had an answer key for each question indicating the main points that we hoped a good answer might address. I am always happy when a candidate gives me an opportunity to tick those points off... and even better when I get to write in a new point because they came up with a good idea that I hadn't anticipated.
I'd never even considered making a corresponding key for things that would lose candidates points. It's theoretically possible that a candidate might say something especially ill-judged that would lower my assessment of them, but I don't think this has happened even once in the interviews I've helped run. Any time I've had to rate a candidate as "unsuitable" it's been because of what they couldn't show me, not because they showed me the wrong things.
[Update: a year and quite a few interviews after I originally wrote this answer, my "not even once" is now "only once". One candidate gave a surprisingly bad answer in the first few minutes of an interview that made it clear they weren't suitable for the job. But the outcome would have been exactly the same if they'd given a non-answer, it just would have taken a little longer.]
If a candidate engages only with the parts of the interview related to technical skills, and greyrocks all the behavioural questions, I just won't have enough to justify a recommendation to hire. You'd have better chances with "here's an example where I made a huge mistake, and this is what I learned from it" than with not answering the question.
(If your prospective employer is really looking to dig up possible negatives on you, that's more likely to be done via background checks, and if applicable security clearance, rather than at interview. It'd be foolish to rely on candidates voluntarily disclosing that kind of deal-breaker.)
How to answer
So, what can you do as an interviewee to help me tick off those boxes?
- Think about the purpose of the question.
What kind of skills is it likely to be assessing? How can your answer help me identify those skills?
If you had a project where everything was great, and all your co-workers were great, and everything just lined up nicely and the whole thing went off without a hitch... I am glad for you, but mostly what this says is "I was lucky", and "being lucky" isn't one of the points I am looking to tick off on my answer key.
If (for instance) the job ad said that we wanted a candidate with good project management skills, that is a strong clue that you should be talking about project management in at least one of your answers.
As a bonus, if you understand the purpose of the question, then you may be able to give a good answer to "tell me about a time when..." questions even when you haven't experienced the exact situation the interviewer is asking about. I discussed this in a recent answer over on Academia SE.
If you're not sure whether you've correctly understood the object of the question, don't be afraid to ask for clarification: "I have a couple of examples I could talk about here, would you rather hear the one about project management, or the one about client relations?" As well as helping you answer the question as intended, this gives you a chance to demonstrate communication skills.
- Focus on factors that are within your control.
Following on from the above, I want to hear about the choices you made that led to a good outcome.
This is where the STAR approach as described in Angelo's answer is very useful. I talked about it at length in this Q&A but in brief, the structure requires you to talk about things you did to make a difference. Don't forget the "action" part of STAR!
- Highlight the parts where you showed initiative.
Sometimes an interviewee will tell me "I did X, and good things happened", but it's not always clear whether doing X was their idea, or whether they were just following a process that somebody else laid out for them. If you're the one who thought up the idea and researched it and then carried it out, be sure to tell me that it's your idea.
- Try to avoid generic answers.
In the most recent interview round, we had one "tell us about a time when..." question where about 20% of our candidates gave minor variations on the same example. They weren't bad answers, but it's a little difficult to stand out when the panel has already heard it. Ask yourself "how many other applicants will tell stories like this?"
Even if your best example is something fairly commonplace like "dealing with an underperforming team member", try to pick out the aspects of that story that your panel won't have heard about in other candidates' responses. As well as making you stand out in their memory, this helps shows you're giving an answer you've thought about, and not just recycling a generic model answer from a careers advice website.