In Java programming, methods and constructors are essential for creating and manipulating objects, each identified by a unique signature comprising the method or constructor name and its input parameters. Errors such as "method X in class Y cannot be applied to given types" and "constructor X in class Y cannot be applied to given types" occur when there is a mismatch in the expected and actual parameters during invocation, highlighting issues like incorrect argument type, order, or number. The Java compiler provides informative error messages to assist in correcting these mismatches. Effective error management is crucial, and tools like Rollbar offer automated error monitoring, making it easier to track, analyze, and manage errors in real-time, enhancing the confidence in deploying production code.