We hosting of a wordpress.org site is not free of charge and I warn you in advance that you get what you pay for. For self-hosting figure an average monthly total cost of about $10 and up generally, but that all depends on storage and bandwidth requirements. You will be responsible for all upgrades, installations, backups and troubleshooting. If you install a theme or a plugin that negatively affects your blog, then you have to figure out what went wrong and fix it.
If you’ve registered your domain through WordPress.com and now want it to point to a blog or website hosted elsewhere then you can do this by updating the nameservers. http://en.support.wordpress.com/domain-mapping/domain-management/