An idea that a party that is protected in some way from risk will act differently than if they didn't have that protection. We encounter moral hazard every day - tenured professors becoming indifferent lecturers, people with theft insurance being less vigilant about where they park, salaried salespeople taking long breaks, and so on.
Moral hazard is usually applied to the insurance industry. Insurance companies worry that by offering payouts to protect against losses from accidents, they may actually encourage risk-taking, which results in them paying more in claims. Insurers fear that a "don't worry, it's insured" attitude leads to policyholders with collision insurance driving recklessly or fire insured homeowners smoking in bed.
The idea of a corporation being too big or too important to fail also represents a moral hazard. If the public and the management of a corporation believe that the company will receive a financial bailout to keep it going, then the management may take more risks in pursuit of profits. Government safety nets create moral hazards that lead to more risk taking, and the fallout from markets with unreasonable risks - meltdowns, crashes, and panics - reinforces the need for more government controls. Consequently, the government feels the need to strengthen these nets through regulations and controls that increase the moral hazard in the future.
The alternative to creating a moral hazard is to simply let corporations fail when they risk too much and let the stronger corporations buy up the wreckage. This theoretical free-market approach should remove any moral hazard. In a true free market, companies would still fail, just as houses burn down whether they're insured or not, but the impact would be minimized. There would be no industry-wide meltdowns because most companies would be more cautious just as most people choose not to smoke in bed whether or not they are insured. In both cases, the risk of burning up is enough to prompt serious second thought on any risk taking behavior.
True free market capitalism doesn't exist, so the taxpayers of many countries are the unwilling insurers for markets. The problem is that insurers profit by selling policies, whereas taxpayers gain little or nothing for footing the bill on the policies and bailouts that create moral hazards.
To learn more about moral hazard, read our article Moral Hazards: A Bump In The Contract Road.
This question was answered by Andrew Beattie.