Companies that continue operation while they await merger or closure, even though they are insolvent and bankrupt. They often continue due to the use of debt.
The corporations tend to only generate enough income to cover the interest payments on the debt. This means they are effectively just pushing forward the fateful day. Ultimately, the cost of servicing the debt increases as more is piled onto the balance sheet.
History
This term originated in Japan as a result of the asset bubble collapse in 1991. Under this scenario, the banks continued to support these zombie corporations rather than allow them to go under. Many feel this helped to lead to the "lost decade" for Japan, although ensuring economic productivity reveals that poor performance was not relegated to the 1990s.
A factor in this is the size of the companies we are dealing with. Governments, particular politicians, are always concerned about jobs. When a major corporation is allowed to fail, it can often take with it 100K jobs, not to mention the fallout with suppliers and other companies associated with the zombie firm.
Here is where we see the "too big to fail" mindset enter.
Risks
There is a great deal of risk to having too many zombie corporations. The process of "kicking the can down the road" does not address the problem.
These companies are extremely vulnerable to changes in interest rates. Since their lifeblood is based upon the repayment of the interest on the debt (servicing the loans), there is nothing done to address the principal. While increasing the amount of debt will be reflected in the cost of servicing, interest rate increases often have a quicker effect. Suddenly, it costs more money to maintain the same amount of debt, a situation often unbearable for firms in this position.
We also can see the larger economy threatened. As these first start to default, the fallout spreads. Markets start to seize up as liquidity is reduced. Jobs are lost, sending unemployment rising. If this happens at a rapid pace and in large enough quantities, we could see recession being the result.
Another issue is that of resource management. As zombie companies are allowed to continue to operate, they serve as a barrier to productive growth. Since they are often misaligned in their use of resources, they detract from those companies that could better utilize them. They employ talented employees who might be more productive elsewhere.
Ultimately, these companies end up bankrupt (or acquired) but not before doing some major damage.