The MissingFormatArgumentException in Java is an unchecked exception that arises when a format specifier lacks a corresponding argument or when an argument index refers to a non-existent argument, causing a runtime error that does not need declaration in a method's throws clause. This exception typically occurs with format specifiers like %d, which require specific argument types, and can be resolved by ensuring all format specifiers in the code have corresponding arguments at the correct indices, as demonstrated by correcting code from using a + operator to a comma to properly pass arguments. Effective error management is crucial in coding, and tools like Rollbar offer automated monitoring and triaging to simplify the process of tracking, analyzing, and resolving Java errors in real-time, thereby providing developers with increased confidence when deploying production code.