Our organization is considering moving to CiviCRM after many years with Z2Systems Neon CRM and in downloading the MySQL database from Neon we note that there are 10 pages of table in the Neon database and only 4 in the Civi template. How do we map Neon to Civi?
Welcome to Civi! I'm not aware of an existing mapping template from Neon to Civi, but there are a number of resources for preparing for such a change. Sanjay's answer (and Jon G's comment) provide a good primer here on StackExchange for getting ready for such a move. The Civi docs also provide an excellent introduction to importing data into Civi.
Another tool you may find useful is API CSV Import GUI extension if you have data outside the standard import fields.