Can anybody register a domain name?
Certainly, if you're entitled to use the tld (top level domain) applied for.
For example, the .com tld is available to anyone at all. All you need is a name that hasn't already been registered, and away you go.
You'll need to go through an accredited registrar, and pay a fee of anywhere between (currently) $9-$22 to register your name for a year.
For other tld's, in particular those that refer to countries, there are often specific conditions for eligibility, which you'll have to demonstrate before being able to register a domain. For example, a .it domain registration requires proof that you're a European union citizen, and is limited to one registration per person. The .ie (irish) domain registration is more stringent again, with companies and individuals having to follow a strict procedure through a limited number of registrars.
Any accredited registrar, though, should be able to guide you quickly and simply through the domain registration process.