From Joomla! Documentation
There are several resources that can help you to get a high-level view of the overall capabilities of Joomla!.
Web sites running Joomla
One way to evaluate a CMS package is to look at Web sites that run on it. It is impossible to say exactly how many Web sites run Joomla, but we do know that over 80 million copies of the software have been downloaded since March 2007. Joomla is the world's most popular full-featured Content Management System (CMS) and powers approximately 3.2% of the largest 2,000,000 Web sites in the world. 
Joomla maintains The Joomla! Community Showcase that allows you to look at a number of sites running on Joomla. Obviously, these only represent a tiny fraction of the total number of Joomla sites worldwide.
A simple demonstration site is available for viewing. This site contains a lot of useful information about Joomla features and its community.
If you are evaluating Joomla for a school, university, or non-profit organization, you might want to look at the Schools and Universities forum.
And, of course, don't forget to include all of the Joomla.org sites as examples of what can be done with Joomla, since they of course are all run on our favourite CMS package.
An important feature of Joomla is the ease - by design - with which its core functionality can be increased by installing 'Extensions'. This extensibility is a major strength of Joomla and a large and active third-party developer community has helped create over 8,000 extensions that are currently available, with more being added daily. Most, like Joomla, are Open Source, whilst the others are offered as commercial software. To get an idea of the variety of extensions available, visit the Joomla Extensions Directory(JED). It should also be noted that the JED does not include templates within its catalogue and there are many thousands of these available for free or commercially.
The developer community is just one part of a larger Joomla community. For example, there are over 700,000 active registered users on the various Joomla forums, with over 150 new participants every day. These forums are very active, with over 1,200 posts per day, and they provide a high level of free support to Joomla users. This support is given voluntarily by other community members. To get an idea of the types of questions answered in the forums, visit the Joomla! General Questions forum or the Joomla! General Questions forum.
An important question when evaluating any software package is the quality of support available. In addition to the support forums, Joomla has a large community of skilled Web professionals and consulting organisations who use Joomla to create and maintain a wide variety of Web sites. Take a look at the Professional Services forum to get a view of this community. There are also a growing number of local Joomla user groups.
Independent market studies
A 2009 report by the independent consulting firm water & stone concluded that "Joomla is the Web’s most popular Open Source content management system". Click here to read the complete report.