One of the biggest challenges in the world is to gracefully take responsibility for a mistake. This is the case for personal errors and organizational ones. How we deal with mistakes in our lives can be the difference between personal happiness, getting that new job, making a relationship work and getting off the phone with IT feeling like you accomplished something.
On a personal level, I always advocate taking responsibility when you should, even when it isn't your fault. If your partner is having a rough day and it's easier for you to take the blame on something inconsequential -- do it. If your little brother forgot to take out the trash or walk the dog and he just can't handle being yelled at -- take the blame and make sure he learns his lesson. You don't want to create a situation where someone cant take care of themselves, but it's also an issue of just moving things along in life. If it's not a big deal and you know it doesn't really matter, then be the hero. Note: this does not apply for huge problems! If your younger sister is getting arrested that she definitely did and you know it -- probably let her take the wrap.
On an organizational level, I believe that pretty much every mistake (within reason) made by an employee should be assigned to their manager and every accomplishment should be shared out from management on down. In other words, a good manager takes responsibility for the team's errors and pushes any accolades for themselves to their team as a whole. A good manager trains and coordinates people in a way that sets them up for success -- and readjusts when that isn't the case.
Most importantly, everyone should learn from a mistake, not just the person who made it. Assigning responsibility isn't anywhere near as important as avoiding the mistake in the future. First, you need to figure out what went wrong and why. Then assess how you can take your current situation and transforming it into where you want to be. Can you skip the mistake the next time by short circuiting the steps that didn't work? Then do it!
Everyone messes up, but not everyone is able to recover from an error without losing their cool. And, at least in my opinion, losing your cool is basically the worst thing you can do after a mistake happens. How does freaking out, yelling, panicking or making any type of scene really help the situation?
My solution? Don't panic and remember that everyone who takes chances also makes mistakes and gets messy. And if you aren't taking risks, then you might want to start -- risk taking is important for personal growth, innovation, creativity and happiness!