How do I connect my domain to my server apps?

Go to your domain provider and configure it to point requests to your server IP address. Use reverse proxy to connect those requests to a specific “app”. Most often NGINX.

I have heard the term DNS a million times but don’t really understand it

The oversimplified version is that it’s “the phone book of the internet”. Users domain requests are sent to it and then it forwards said requests to the appropriate IP address.

I’d like to let family access my media server, are there any considerations I need to make?

Are you hosting on a VPS or a home server?

How can I use one domain to access multiple services on my server?

Subdomains

Do I need to pay extra for subdomains?

No.