To be able to install and run Joomla!, you'll need hosting space and a few tools. Here's your shopping list.
First of all, you'll need hosting space; a place on a web server where you can set up your site to make it accessible to anyone with Internet access. Your hosting account should support the PHP scripting language (as Joomla! is written in PHP); it should support the type of database that Joomla! uses, called MySQL; and it has to run the Apache server software. Specifically, these are the system requirements for Joomla! 1.5:
PHP 4.3.10 or above
MySQL 3.23 x or above
Apache 1.3.x or above
You shouldn't have any difficulty finding hosting accounts that meet these system requirements. If you're not sure, any hosting provider should be able to tell you if they support Joomla!.
You can find detailed system requirements at the Joomla! official help site: http://help.joomla.org/content/view/1938/302.