Host names are optional - you can always connect to a game using its IP address. But it’s nice to have a host name (like yourmush.somemuhost.com) so people can remember your game address more easily.
Most VPS servers don’t include a host name. If you want a name, you’ll have to set it up.
You can request an AresMUSH sub-domain (yourmush.aresmush.com) by contacting me directly. AresMUSH sub-domains are given out at my sole discretion, and at no charge.
You can also register a custom domain name (yourmush.com) for just a few dollars a year through sites like NameCheap. Once you have a domain, you can point it at the IP address of your droplet. The steps will vary depending on the domain provider.