I want to approach this from a different perspective. I think the other answers have covered well what you do when there is a major mistake particulaly when complex processes are involved.
However, not all mistakes are process driven. People still do the work and people make mistakes that are based on several things:
- Incompetence (or lack of knowledge about a particular issue not just
- Personal stress unrelated to work(Divorce, death in the family, etc)
These are often not the kind of mistakes that drive the need for a massive effort to do post-mortem analysis. But they do drive the need to deal with the individual who is making more than the occasional mistake.
So let's deal with some of those possibilities. I'm going to lump incompetence and carelessness together as the process to deal with them is similar. First, explain to the person, what was wrong, why it was wrong and how to do it correctly. Then make the person make the fix. If they learn from this and don't continue to make the same mistake, then that's all that is needed.
But people being people, this won't fix everyone. Some people will need private counseling about their problems and why they keep making them and may need to be put formally on a plan to fix their lack of skill or caring about the quality of their work. This needs to be a plan with consequences. If the person doesn't shape up after giving several chances to fix the problems they have, then you may need to let them go. And they need to know this is a possibility if they don't fix what is wrong. It should never come as a surprise if you are getting let go for lack of performance. But that first talk shouldn't be the kiss of death either, if the person straightens up and improves, it should be possible to come off the performance improvement plan and be treated the same as the other employees.
Now in the processing of the private discussions to find out why the mistakes are happening, you may find that distraction, exhaustion, or personal stress are involved. If distraction is causing a problem, then as a manager, you need to fix whatever is causing the distraction. That might mean not expecting the person to multitask, it might mean moving to a quieter location, it might mean having someone else handle technical questions or help desk calls or what ever is the thing that keeps pulling the person off task. Then see if performance improves.
If exhaustion is involved (and is work-related), the manager probably is the one who made the mistake and should have noticed that the person was working too many hours. If the person is working more than 8 hours in a day, send him home and tell him not to come back until he has slept a full night. Then stop planning tasks so that they require more than 40 hours a week. Employee exhaustion is often the result of management errors. So take the blame and fix the problem. Now sometimes an employee is exhausted due to other issues outside of work and those will come under the next paragragh discussing personal stresses.
If the person tells you of outside reasons why he is making mistakes, then you need to assess those reasons and come up with a plan to mitigate them. Sometimes this means a short-term lightening of the load (as was done for me when my beloved died), sometimes it might mean moving to a less stressful position permanently (in the case of someone who will be under the stress for a long time, such as when somone has become a caregiver), it may mean checking their work more frequently. It might mean a change of schedule. It might mean putting him on medical leave or short term disability or part-time hours. It may mean telling him to quit that second job or requiring they get medical treatment for depression or insomnia or a whole host of other things.
Anytime you need to go one step up from just having the talk about the mistake and how to fix it or the identification of a personal stressor that is affecting work that can't be fixed by a simple short-term rearrangement of work, you may need to get HR involved before you deal with the person directly. They are aware both of legal issues you may not be aware of and of possible resources (as well as ways and means as to how to do somthing like put someone on short-term disability) and company policies you need to follow. They are the manager's friend when there is serious performance problem that needs to be addressed.