Whose failure? If it is a personal failure then be frank about it. Along with owning up to it show evidence of:
1. handling the mistake competently, or
2. handling a unforeseeable environmental cause of failure competently.
Trying to deflect the blame to an abstract source, such as saying, "Mistakes were made." is popular advice because often failures if attributed to leaders will end their tenure at a company. However I don't agree with this advice on two fronts.
Front one: I don't believe leaders are fit to lead if they are willing to work in a culture that penalizes executives who make intelligent mistakes or a project fails due to unforeseeable events, and rewards executives who employ "smoke and mirrors" to make it impossible to track down the cause of an failure and learn from it.
Front two: When leaders own up to failure and showing the failure was not the result of carelessness but happened despite intelligent forethought, and that they are now competently addressing the issue they create a positive culture that enhances organizational success. Here's how they show well thought out plans that are fail are not penalized, and they set an example of how to act when something fails.
If you're addressing an employee's failure then seek to understand the cause. If it was a well thought out project that went awry, and the people, or person, involved is learning from it then don't do anything other than congratulate or reward the individual(s) for a smartly developed plan that just didn't pan out. Now if the employee failed due to what a competent and reasonable employee should have recognized would lead to failure it is prudent to ask:
Was it due to ignorance - in that case retraining is necessary as well as looking at hiring and promoting practices, because a person shouldn't be in a position where she is incompetent.
Was it due to laziness? In this case the person ought to be counseled and informed of consequences of change not occurring.
Was the failure due to rebellion? If the person knew the undertaking was contrary to acceptable practice and went ahead with it - fire the person.