This is a tricky one. Isn't it? How do you tell the evaluator about your failure & weakness without making it look bad on your application? Blunders are all too common on this one. Let us discuss how we can safely negotiate this question with an attempt to possibly score few additional points.

First, do not duck the question. The last impression that you want to give is that you can't take tough questions. That will be unacceptable to the adcom. Similarly, there are easy answers which just wont do, ex. 'I am a perfectionist and that at times works against me'. This is just as bad or maybe even worse. Everybody, including your evaluator, is rational to believe that no person is perfect. An answer like this will make him/her question your credibility. And that is definitely not something you want to happen.

Now, find an authentic imperfection or a situation where, you would like to have acted differently. When you talk about these situations, the overall sound has to be very positive. DO NOT blame others (boss/colleagues/clients) for the incidence; take it on your chin. The most important part is to demonstrate how you have worked to correct or improve on it. Talk about the lessons learned. If possible site the example of a similar situation later where you did the right thing. That will demonstrate that you are an authentic person, can face difficult questions, and is ready to learn. The idea is to see every mistake as a learning opportunity.

However, before you go overboard with finding that authentic imperfection, please hold your guns. Some weaknesses are unpardonable. Example, I am lazy, I can't tolerate dumb people, I stole other people's credit during a project, etc. All and all, this is a section where you should largely look to play safe. There is typically not too much to gain but a lot to lose in case of a bad answer. An example of a 'good' weakness is 'I am not good with technology, and I see that as a major weakness in this day & age (for somebody who is not from core technology space)'. Again, the most important part is to mention the learning. And in case of weakness, the effective steps to address the weakness.

And lastly, keep weakness/failure answers handy for your interviews, irrespective of whether your application essays/profile ask for it or not. You don't want to be guessing at the interview.
Personally, this was one mistake which almost cost me my own admit few years back. Just a little awareness will see you safe and sound.

Anshul