In order for people in the outside world to locate your servers under the domain name of your choice, whether for web, FTP, or email delivery, you will have to register the domain name for insertion into the relevant top level domain database.
Exercise some simple prudence in choosing your domain name. Certain words or phrases may be forbidden on the grounds of community standards, or may be offensive to visitors whose language or slang differs from that of your region. Domain names can contain only the 26 letters of the Roman alphabet (without accents), the hyphen (though not at the beginning or end of the name), and the 10 digits. Domain names are not case-sensitive, and can be at least 26 characters long (this limit is subject to change). Be careful not to register a name which you can reasonably have been expected to know infringes on the trademarks of an existing company, the courts are not kind to cybersquatters. Some information on the circumstances under which your poorly-chosen domain name might be stripped from your control are available in this Uniform Domain Name Dispute Resolution Policy.
There are many companies which register names in the ``.com'', ``.net'', and ``.org'' top level domains. For a current list, check the list of accredited registrars.
To register a name under a country top level domain, such as a ``.ca'', ``.de'', ``.uk'', etc., check with the appropriate authority, which can be located in the Country Code Top-Level Domains database.
Typically, you have to provide the registrar with contact information, primary and secondary DNS IP numbers, a change request validation scheme (you wouldn't want just anybody changing your domain for you), and money in the form of an annual fee. If you're not comfortable with the change request validation schemes offered by a registrar, let them know that you're not willing to use the service until they address your security concerns.