This might be very obvious to some, but about 70% of the clients I work with do not know the difference and spend money on something they could be getting for free! If you have a blog or would like a blog it is essential that you know the difference between WordPress, wordpress.com and wordpress.org. The difference is not communicated at all, especially on wordpress.com and I really hate them for this.
First of all, let's have a look at what WordPress is. WordPress is an open source project and in the creator's words from the about page: "it also means you are free to use it for anything from your cat's home page to a Fortune 5 web site without paying anyone a license fee". This means WordPress is completely, totally and utterly free. You can show ads, you can modify the code, you can sell it, share it, whatever you want.
WordPress.org is the website for this open source project. It's actually a really great place to learn about WordPress and its usage, even if you are an advanced programmer it can give you a lot of help! You can also download the latest versions for yourself.
Now then, how do you use WordPress? To use it, you need your own domain name and hosting. You download the zip file, extract it on your server and follow the install manual you can find on wordpress.org. You will need to set up some databases, but this is really not a big deal, and even if you've never heard about databases and php you should be able to find your way around in an hour or so. If you've dealt with databases before installing WordPress takes 5 minutes. From this point on you have your blog and you can do whatever you want with it.
Some web hosting and server management solutions come with options to set up a WordPress blog automatically in the background. This makes things even easier, as it does away with most of the technical process (downloading, installation, initial setup).
So where does wordpress.com come into all this? WordPress.com is a site exactly like Blogger, it allows you to have a blog if you do not have a domain. A domain + hosting costs about $100 a year if you have a good host and if you don't have the cash to spare you can go for a free blog. In this case you can head over to wordpress.com and get a free blog, but you will have a domain in the form of myblog.wordpress.com.
Since WordPress the software is free and openly usable wordpress.com uses WordPress as the software behind its blogging. However, they charge you for being able to modify your CSS and being able to put Adsense ads on the site and so on. The thing I hate is that they charge you for adding a domain as well, so if you have myblog.wordpress.com and you buy myblog.com you will need to pay them to be able to redirect your domain.
In any case, what you need to know is that if you have your own domain and hosting you do not need to go through WordPress.com, they are just a third party facilitator for people who do not own domains. If you do own your own domain however, WordPress is completely free in terms of money, and completely free in terms of usage and features.