Take responsibility for whatever you did and understand the possible impact. 

Give yourself room to be wrong, and don't defend yourself in hindsight. Defensiveness is your number one enemy.

Pinpoint how did the mistake happen and where things went wrong. Then ask yourself what you can do to fix the problem.

Assess if you need help or you can fix it on your own. Don't try to sweep it under the rug--that could backfire.

Communicate the mistake to those affected or to those who can help, in ways that appeal to people's humanity.

Be sincere. Show that you care to fix it.  Be honest about your struggles.

Expect to make public apologies.

Use them to cure yourself and your whole company of being defensive. Of course, do whatever you can to not repeat a failure.

Taking responsibility means sacrificing your pride, not pumping your ego. It also means showing trust in people to help you.