CiviCRM hosts several demo sites - there is at least one for each of the supported Content Management Systems (CMS) - Drupal, Joomla, and WordPress. The demo sites present a working copy of the latest stable version of CiviCRM with sample data. You can use them to play around with CiviCRM, but be aware that they are publicly viewable so you shouldn't enter personal data.
A list of the available demos can be found at http://demo.civicrm.org/
The demo sites may be configured strangely, missing functionality or appear in different languages, as other people may be experimenting with them at the same time as you.
The demos have some limitations - you can't send emails from them, you can't set permissions for Drupal users or do a full exploration of online payment options.
If you are having trouble working on a demo site, contact the CiviCRM core team via the forum or IRC. If you want a more controlled environment for exploring CiviCRM, install your own test site.
You can download and set up a local version of CiviCRM, that is a version that is stored on your local computer rather than on a server on the Internet. You'll still access it through a browser, but it will only be visible on your computer. The advantage of a test installation is that you can configure CiviCRM, and experiment with your data.
Up-to-date information for installations (including troubleshooting tips) is maintained at http://wiki.civicrm.org/confluence/display/CRMDOC/Installation+and+Upgrades.