CiviCRM is known to work with Joomla versions 2.5.x, and 3.x. The most current versions of CiviCRM will not work or install on versions 1.x or 1.5.x.
CiviCRM is NOT well-suited for inexpensive shared-hosting environments, however, many users have deployed CiviCRM successfully on minimal hardware.
CiviCRM is installed as a Joomla component, and makes several menu item types available, as well as php-cli/web-api interface by which site admin may schedule and run CiviCRM tasks, such as scheduled mailings, batch geocoding of contact address data, as well as updates of several types to various contact, and member data based on user participation in a wide variety of activities, and contributions. CiviCRM also has a robust membership/constituent data-model, allowing for many variations of activities, contributions, votes, pledges and mailing list subscriptions. CiviCRM also features interfaces to popular payment gateways, for paid, or "requires-donation" type activities and events.
CiviCRM is the ONLY "free" CRM extension for Joomla which offers so much functionality in one component, however, inexperienced site admins may find it difficult to correctly configure and deploy.