Parameters are simple input to JasperReports. You have to define parameters in the JasperReport before using them. You can display the value of the variable, you can use it as part of boolean expressions and you can even pass it to subreports. This can be an input to SQL query.
Fields are simple variable definitions. You can think of these as instance variables of the datasource object thats passed in to the report or they can be key names if the datasource is a Map. If you configure JasperReport to create the dataset based on SQL, then Fields are the column names of the ResultSet. You will use Fields to display the resultset of an executed SQL query.
Variables are another kind of variables that live within Jasper Report, they are not inputs. They are used to calculate sum or average of certain Field (defined above). You can perform many other predefined calculation functions on the Fields using Variables.