Java runtime errors, occurring during program execution, can be categorized into runtime errors and runtime exceptions, with the latter further divided into checked and unchecked exceptions. The ExceptionInInitializerError, a specific runtime error, arises when an unchecked exception occurs during the evaluation of static initializers or static variable assignments, causing the Java Virtual Machine to terminate the program. This error acts as a wrapper for the underlying exception, preventing the program from continuing until the issue is resolved. Handling such errors involves ensuring static initializers and variable assignments do not throw unchecked exceptions, and implementing error-handling mechanisms like null guards or try-catch blocks. Tools like Rollbar can assist in managing these errors by automating error monitoring and triaging, thereby enhancing the confidence in deploying production code.