There are two kinds of Wordpress blogs: self-hosted - Wordpress.org - and Wordpress.com, which is hosted by Wordpress. Both are free but have different advantages.
If you want complete and total control over your blog, download Wordpress from
http://www.wordpress.org - it's completely free. Then go find yourself a host that allows access to your own MySQL database and allows PHP. There are several out there, some that may not even force banner ads on your blog. Installing Wordpress is very easy, and setting it up generically is easy. You can theme your blog however you can imagine, provided your host allows the PHP you choose - some will block certain functions for safety reasons.