Wiktionary
n. A problem or situation that occurs only at an extreme (maximum or minimum) operating parameter.
Wikipedia
An edge case is a problem or situation that occurs only at an extreme (maximum or minimum) operating parameter. For example, a stereo speaker might noticeably distort audio when played at its maximum rated volume, even in the absence of other extreme settings or conditions.
An edge case can be expected or unexpected. In engineering, the process of planning for and gracefully addressing edge cases can be a significant task, and one that may be overlooked or underestimated. Non-trivial edge cases can result in a failure of the object being engineered that may not have been imagined during the design phase or anticipated as possible during normal use. For this reason, attempts to formalize good engineering practices often incorporate information about dealing with edge cases.