This dialog lets you to tweak the logging level of java.util.logger.Logger in the server process. This is often useful for getting more information out of Jenkins for trouble-shooting.
In Java, loggers are identified by their names, such as "hudson.model.Project", which designates the name of the class where the logging message comes from. Loggers are also organized into a hierarchy in the same way classes are organized into packages, so "hudson.model" and "hudson" are the ancestor loggers for "hudson.model.Project".
Logging level is a property for individual logger, but most loggers are by default configured to inherit its logging level from the parent. This means that except the loggers where the logging levels are configured, you can affect the logging level of a whole package by simplying setting the logging level for that package.
Refer to Jenkins source code or the mailing list archive for what logger corresponds to what feature.
See Java Logging Overview for more about the java.util.logging package.