You can't install software on a domain.
A domain is just a name that gets resolved to a server (which you have to pay separately in most cases); This happens via DNS records.
So basically you need to buy a server (or host one by yourself which I don't recommend, especially for beginners) and a domain.
You set-up your domain to use the nameservers (NS) of your server provider (or the IP directly, but if your server provider has nameservers it's recommended to use them). In most cases you just tell your server provider which domain to resolve to which server and he'll generate a basic set of DNS records. After a period of 4 to 24 hours (in most cases) the connection between domain name and server will be established. In the mean time you can set-up anything on your server like NodeBB (you can test by using the static IP address of your server instead of domain name).
For a server provider a common choice is digitalocean which offers pricing from USD5 per month. You can get a USD10 credit using this referral link.