WordPress.com provides free blogs and free hosting for them. WordPress.com does not accept domain name transfers but mapping to an existing domain can be done.
Domain mapping of an existing domain URL that you already own and purchased elsewhere is not done free of charge. You will have to renew the domain name purchase where you made it and the domain mapping purchase here every year.
Important Notes Before Upgrading http://en.support.wordpress.com/domains/#important-notes-before-upgrading
Mapping an existing domain costs $13. annually. In order to map to a domain one must have an underlying .wordpress.com subdomain blog to map from. If you have registered a .wordpress.com blog you log in as Admin under the exact same username account that registered the blog and purchase a domain name by purchasing an annually renewable domain mapping upgrade for the blog.
See here for mapping an existing domain that you already own http://en.support.wordpress.com/domains/map-existing-domain/
Dashboard > Store > My Upgrades is where you do the primary blog set up knowing it can take between 24 - 72 hours for domain name propagation to take place throughout the internet. You can view the DNS changes here > http://www.whatsmydns.net/ What's important during that stage is to be patient.