Why you need to go for self installed wordpress blog and not the free one at wordpress.com
WordPress is available as a free blogging platform (CMS or Content Management System) via wordpress.com. There are other platforms like blogger.com (from Google) but those are blogging platforms where you can create blogs within their domains like yourblog.wordpress.com and it is hosted in their sites.
Though you can redirect your domain name, you will not be able to move the files to your hosting or make complete customizations to your blog. your customizations are limited and even the limited ones are available via paid plans. The terms and conditions and options to use your blog to the fullest is limited. Thats why wordpress has its installable as a open source version via wordpress.org where you can download the wordpress CMS and install in your site under your hosting. You are 100% free to make any changes or customizations to “your” blog as you wish.
Also this usage of WordPress in your hosting and domain makes it your property built on your land.