No problem -- it IS a bit confusing. WordPress.ORG only provides the WP core software, free themes and plugins that are on this site. Your registration/user name here is only for this site/these forums - unlrelated to WordPress.COM or hostgator (or your site there). The installation is done on your host - which you've already done :). And if you didn't put anything on your WordPress.COM site, no, you don't need to move anything (i.e. content from your site) from WordPress.COM. If you registered your domain name through WordPress.COM, you DO need to change the nameservers to hostgator. This explains how to do that:
http://en.support.wordpress.com/domains/change-name-servers/
That can take up to 72 hours, but hostgator may be able to give you a temporary domain to use in the meantime - ask them about that.