A moral hazard is when the consequences are no longer tied to a decision. The classic example is when someone buys insurance. Because the negative consequences are covered, you will be more likely to take a risky action. For example, if you have health insurance, you're more likely to go horseback riding.
Another way to look at moral hazard is to invert the idea. Nassim Taleb talks about "Skin in the Game." In other words, when you fail, you suffer the consequences. Taleb claims that "Skin in the game is necessary for fairness, commercial efficiency, and risk management."
Why is moral hazard important? Moral hazard distorts decision making.
It's interesting that on a personal level it seems like moral hazard is a good thing. It's good to be punished less than you should be. But on a society level, moral hazard leads to the wrong actions being taken.
Even on a personal level, the concept is useful. You want any partners you have to have skin in the game. You don't want to work with anyone who won't have any downside if they fail. And if you think about it, you will be more motivated if you have skin in the game. While it might seem better to remove downside, with less downside perhaps you have less upside, or less probability that the upside will happen.