A genuine issue of material fact is a circumstance in which two parties in a lawsuit have a dispute about certain facts that are important to the case. In order to qualify as "genuine," the party that is challenging the facts must be able to demonstrate that there is a legitimate reason to believe that the fact is not true.
Genuine issues of material fact can arise when one party in a suit seriously doubts the credibility of a fact being presented in the case. A summary judgment is often needed to resolve these issues.
An example of a genuine issue of material fact would be if one party claims that there was a brand new model Corvette at the scene of the crime, yet that model Corvette had not even been released at the date that the crime took place.