Drupal and Joomla are content management systems.
The sites run on back end databases.
I haven't tried Joomla, but I do have limited experience with Drupal and it's your worst nightmare to get it to do what you want it to.
Be prepare for a very steep learning curve.
At the very least, you need a Linux or Windows server that supports apache, mysql and php (a so called lamp server). If you're using a hosting service, check what they support.
A knowledge of php would also be helpful, otherwise you are stuck with existing themes. Not that they are bad, just limited in what you can design within.