.jrxml is a human readable XML file that contains the report template i.e. report structure and its formatting rules.
.jasper is the compiled report template i.e. compiled .jrxml file. You use this file as the template argument in the JasperReports API.
.jrprint is a serialized JasperPrint object i.e. an actual report instance i.e. a template that has been filled with data. This file can be deserialized back into a JasperPrint object.
.jrpxml is a human readable XML represenatation of a JasperPrint object i.e. an XML version of a template that has been filled with data. This file can be unmarshalled back into a JasperPrint object.