Here's how it's described on the Joomla! official site at http://www.joomla.org/about-joomla.html ( I highly suggest you read this one page article.)
What is Joomla?
Joomla is an award-winning content management system (CMS), which enables you to build Web sites and powerful online applications. Many aspects, including its ease-of-use and extensibility, have made Joomla the most popular Web site software available. Best of all, Joomla is an open source solution that is freely available to everyone.
I think a visit to joomla.org will answer most of your questions, there are a few questions that people ask that won't be covered there.
Q. If Joomla is free why do I have to pay to get a Joomla site from a developer?
A. Simple. You don't have to pay. You just have to study the installation manuals and learn how to use it. If you don't want to study - you need to pay someone to do a proper installation for you.
Q. Why do all the developers want me to host my site with them since Joomla is so open and free? Can't I host it anywhere?
A. Technically you can. But Joomla requires specific settings, software versions and graphic libraries. (Read this 10 Stupidest Administrator Tricks)
It also requires permissions to be set properly. Again, if you know how to do it, you can host it anywhere. However, someone who specializes in Joomla development knows what the pitfalls and roadblocks are and will save you countless hours of frustration and work. Answers are available, but you'll have to find them on forums and over the internet, and that takes a lot of reading. Not all servers are set up the same or optimized to handle Joomla. A developer generally wants to work on a server they are familiar with. There will be constant updates and a hosting company that specializes in Joomla hosting will be able to keep all their customer sites up to date more easily. Mass market hosters like GoDaddy will not give you technical support on Joomla itself. The hosting is cheap, but you're on your own when it comes to your website with a budget host. I'll build you're Joomla site on any server. However it will cost you more because it will take me longer than on my server where I know what to expect, and then I'll have to charge you for the fixes and updates when it breaks.
Q. Why does it cost to transition the old website to a new Joomla site?
A. Your old site is full of content. It all has to be copied and reformatted to fit the Joomla framework. That means conversion isn't much different than building a new site. Most likely you will have all the content and pictures, but they have to be arranged and integrated. You may also have some type of programming or function like a slide show, data base, or link directory that can't be moved with a simple copy and paste. You'll need to install additional modules and troubleshoot them. Sometimes it's easier just to do a new site.
Q. Why should I change? The old site is working.
A. Yes, but for how long? Microsoft and every other software developer continues to come out with new versions of products. Security holes are constantly being patched, and what worked yesterday might not work today. You may never find anyone to fix the old software and you'll be forced to do a major overhaul. Joomla, on the other hand, having a developer community of over 200,000, and being in existence since year 2000, has evolved steadily with regualar upgrades and migration paths. They're a lot like oil changes. If you keep up with them everything lasts longer.
Q. I repeat, the old site is working. Why change?
A. You might have noticed the rise of social networking in the last couple of years. Old programming just isn't developed to take advantage of the newest advances. You may be able to use a good old static web page for a while longer, but in the not too distant future, the people who know how to make use of social media will bury you. Joomla has native blogging capabilities and people have built on community components so you can turn your Joomla website into something that rivals Facebook.
If you are using a website that is more than two years old, uses static pages, relies on CGI scripts or was created with a site-builder, AND you are seriously trying to do business on the internet, you should be looking into Joomla now.