A domain name is an address for the internet that identifies the name of a website.
The registration is made by associating an IP address to the chosen domain name, or rather a numeric string that allows you to reach the website from the internet.
In other words the domain name www.aruba.it is simply the alphabetic translation of the IP address 18.104.22.168, therefore, by typing the name of the website (www.aruba.it) or the IP address (22.214.171.124) on a browser you will reach the same website.
The association between the mnemonic (www.aruba.it) and the IP address (126.96.36.199) must be univocal so that whoever tries to reach www.aruba.it, they reach the same website, regardless of where they are, the provider used for the connection etc. as it is for telephone numbers: if you call a specific telephone number from different places, you always reach the same person.
For each TLD - Top Level Domain (the last part of the domain name e.g.: .it, .com, .eu) there is a Registration Authority that manages the database of registered domain names; for example the .it TLD is managed by the .it Registry. To guarantee that the association between the IP address and the domain name is univocal, there is a database, called Whois, which contains all the registered domain names.
Therefore to register a domain name consists in following the procedure in order to place your domain name in the database, guaranteeing that such domain name is for your use only.