Which files do I need to send the other party for them to copy the files to a server and then make use of the tables.

Its a non-transaction table so i guess MyISAM.

1) Make a dump of your database
2) transfer it to the other computer
3) Import the dump

For 1 and 3, 2 ways of soing it:
If you have a shell access:

mysqldump --create-options your_db_name>my_db_dump.sql

You can specify which tables you want to export too:

mysqldump --create-options your_db_name tbl1 tbl2 tbl3>my_db_dump.sql

And to restore it:

mysql -u db_user -p db_name < my_db_dump.sql

If you don't have a shell access, then use whatever tool you have access to.
Mostly, it will be phpMyAdmin, which have an export function.
http://drupal.org/node/81993

To reimport the datas, simply upload the SQL file to be executed.

http://uk2.php.net/manual/en/book.ftp.php < FTP function in PHP.