If you have been publishing a blog under WordPress.com then you already know that it is not hosted under your own domain name. The link to your blog must be something like http://yourblog.wordpress.com. For running an elementary blog this is a good option but once your blog has evolved and you are interested in more serious blogging including revenue options, you will need to convert your WordPress.com blog into a WordPress.org blog. No, it does not mean that now the link to your blog will be http://yourblog.wordpress.org, it means you will be downloading the WordPress setup files from WordPress.org and then uploading them to your own server in order to setup the blog under your own domain, http://yourblog.com. See the difference?
The benefit of publishing a blog under your own domain is you can do whatever you want to do with it. You can have whichever theme and layout you want. You are not obliged to adhere to terms and policies of WordPress.com. You can publish ads on your blog and earn revenue. Your own blog helps you in branding and promotion. If you are running a business blog, something like http://yourbusinesswebsite.com/blog/ it can also help you improve your search engine rankings tremendously. The biggest benefit of running a blog under your own domain is that you are in total control of your content. If tomorrow WordPress.com goes down, you will have your entire content with you. Importing and converting has become so easy these days that even if there is no support for WordPress.org set of files one day, you will still be able to run your blog using another platform but having the same content and layout.
The first thing you need to do in order to be able to use WordPress.org is to have your own domain name. You can register a new domain, or you can upload the setup files under a subdomain of an existing website. The choice is yours. You should also have direct access to your MySQL server because you will need authentication permissions in order to set up and run the blog.
After this, go to WordPress.org and download the setup files in a folder on your local drive. You may need to unzip the setup files. Once you have done that, using your preferred FTP client program, you will need to login to your web hosting FTP account and then upload all the setup files there.
Setting a WordPress is quite easy and it is mostly self-driven. After uploading the files you simply have to visit the URL and the setup begins on its own. Keep your MySQL server login details handy because you will need them as a set of proceeds.
Once the setup is done, you will be given a temporary password that you can use to login to your dashboard and once you are there, first of all change your password.
Then you need to login to your WordPress.com account. You will need to export all your blog posts into an XML backup file using the inbuilt interface of the dashboard. Since WordPress.com also uses the core WordPress setup files, there is not much difference in the dashboard interface. The entire procedure of first exporting your content from WordPress.com and then importing it into your own blog setup is explained in this blog post with all the necessary visuals.