Additional to your aternos.me server address, you can connect your own custom domain to your Aternos server.
This requires you to already own the domain you want to use. It is not possible to buy/register new domains via Aternos.
There are also sites that allow you to register a domain name for free.
Setting up a subdomain of your domain
To connect a subdomain of your domain to your server, create two NS records, one for each of the Aternos nameservers, in the DNS settings of your domain. Set the 'Name' field to your desired subdomain in both of them.
The result should look something like this:
Setting up your main domain
It is also possible to connect your main domain to your server; this will, however, prevent you from creating any additional DNS records or subdomains for it.
To do so, you have to change the nameserver settings on the website of your registrar (the website you bought/registered your domain on) to point your domain to the Aternos nameservers.
Connecting your domain to your server
After setting up your domain/subdomain correctly, you can then add it to the domain list of your Aternos server (https://aternos.org/domains/). DNS and nameserver changes can take up to 24h to take effect, so this step might not be possible immediately after changing your domain settings.
Creating a free domain for your server
If you don't own a domain for your server, there are sites that allow you to register one for free, e.g. https://freenom.com.