The word "core" in Joomla! pertains to the distributed files which are needed to create and administrate a Joomla CMS powered web site. These files can be downloaded from the Joomla website at http://www.joomla.org/download.html. The Joomla "core" also contains some basic functionality to get new Joomla installations working quickly and easily. Included are the user manager, article manager, weblink manager, category manager, contact manager, and menu manager. There is also a template manager with a few basic templates to power the front-end (website/user) view, a module manager with basic modules, plugin manager with basic plugins, and a few other "out of the box" extensions to extend the functionality of a basic Joomla installation. These core's extensions should not be confused with extensions which are available for downloading from the JED (Joomla! Extension Directory).

Reference: http://docs.joomla.org/Glossary

A template is a type of Joomla! extension that changes the way your site looks.

There are two types of templates used by the Joomla! CMS: Front-end Templates and Back-end Templates. The Front-end Template controls the way your website is presented to the user viewing the website's content. The Back-end Template controls the way your website's administrative tasks are presented for controlling management functions by a Joomla! Administrator. These would include common tasks such as: user, menu, article, category, module, component, plugin and template management.

Reference: http://docs.joomla.org/Glossary