Presuming that you have correctly setup an A record pointing your domain name to the droplet's IP address, there are two files you'll also want to edit on the server in order to get up and running using the domain name rather than the IP address.
First, edit your Nginx configuration at
/etc/nginx/sites-available/ghost and change the current
my-ghost-blog.com to your domain name.
Next, edit your Ghost blog's configuration file at
/var/www/ghost/config.js and update the production url as well:
Finally, restart both for the changes to take effect:
service ghost restart
service nginx restart
Check out this tutorial for all the information on using the Ghost One-Click app:
DigitalOcean's one-click "application" installation allows you to create a droplet with Ghost pre-installed, running and ready to go in just a few short minutes! The instructions below will take you from zero to blog.