Final answer:
A run-time exception that signals an illegal cast operation is known as a ClassCastException. This occurs when an object is cast to an incompatible type, resulting in a runtime error.
Step-by-step explanation:
A run-time exception thrown to signal an attempt to cast an object to a class of which it is not an instance is called a ClassCastException. This exception occurs when code attempts to cast an object to a type that it does not inherit from, violating the object's type safety. When the Java Virtual Machine (JVM) detects this invalid cast, a ClassCastException is thrown, indicating that the cast operation is not legally allowed at runtime.