Moral fiber is the capacity to do what is right, no matter what the circumstance. A business person with strong moral fiber, for example, would decline a bribe, and put his self interests aside for the benefit of the company and its shareholders. A parent with strong moral fiber does whatever it takes to help their kids succeed, even if it means they have to make sacrifices. A person with moral fiber does not do things to intentionally harm others, and when given the opportunity, does things to help the less fortunate.
When Abraham Lincoln pushed to free slaves despite hostility from his own political party, he showed great moral fiber.
Bernie Madoff was severely lacking moral fiber when he took billions from investors.
A parent that would rather drink and get high than pay rent or electric bills has no moral fiber.