If you ever change your web host (the actual physical location of your website files), your domain name goes with you. Your regular visitors or customers who knew your site name as (say) www.thesitewizard.com would not have to be informed about a change of web address (also known as "URL"), since as far as they are concerned, the site is still at the same place. They would simply type your domain name and they'd be transparently brought to your new location.
The character set allowed in the Domain Name System is based on ASCII and does not allow the representation of names and words of many languages in their native scripts or alphabets. ICANN approved the Internationalized domain name (IDNA) system, which maps Unicode strings used in application user interfaces into the valid DNS character set by an encoding called Punycode. For example, københavn.eu is mapped to xn--kbenhavn-54a.eu. Many registries have adopted IDNA.
If you do not have a web host, you can always allow the registrar to park your domain name at a temporary website specially set up for you. This way you can quickly secure your domain name before it's too late and still take your time to set up the other aspects of your site. As far as I can tell, most registrars (or probably all) automatically park your domain by default whether you ask them to or not, so if this is your situation, you probably don't have to do anything special to get it done. Some of those registrars also provide you with a free email address at your own domain name while it is parked at their site, like sales@example.com (where "example.com" is your domain).

For the most part, you can pick whichever TLD you prefer, though a few have restrictions -- you can't buy a .edu domain unless you run an accredited post-secondary school. Many, but not all, country-specific domains (.us, .uk, .jp) are restricted to citizens of the country in question. Unrestricted country domains are often used for "domain hacks," where the extension becomes part of the title, such as YouTube's alternate address, "youtu.be."
Not every registrar sells every top-level domain, so if you want an uncommon TLD like .pizza, you might have to try a few registrars. Unusual TLDs often cost more than a standard .com address, but provide built-in memorability and give you more name options, since far fewer .pizza addresses are already taken. You can check name availability at any registrar that offers the TLD you want.
ICANN publishes the complete list of TLD registries and domain name registrars. Registrant information associated with domain names is maintained in an online database accessible with the WHOIS protocol. For most of the 250 country code top-level domains (ccTLDs), the domain registries maintain the WHOIS (Registrant, name servers, expiration dates, etc.) information.
If you do not have a web host, you can always allow the registrar to park your domain name at a temporary website specially set up for you. This way you can quickly secure your domain name before it's too late and still take your time to set up the other aspects of your site. As far as I can tell, most registrars (or probably all) automatically park your domain by default whether you ask them to or not, so if this is your situation, you probably don't have to do anything special to get it done. Some of those registrars also provide you with a free email address at your own domain name while it is parked at their site, like sales@example.com (where "example.com" is your domain).
If you're having trouble finding a domain name (whether because of crowding or cyber-squatters), check for a help facility on each registrar's site. Domain registrars typically house search engines that return a listing of available names similar to the one you want. When you search for a domain name at Namecheap, for example, you get both the status of that name and a list of suffixes available for that name. Maybe [Sitename].com isn't available, but [Sitename].biz or .org is.
Some domain names are only available at auction. For now, we’ve chosen to feature buy it now domain names because they are often available immediately. We are working to improve our search engine to make it easier for advanced users to choose whether they want to include domains at auction (or domains with hyphens and numbers) in the search results.
You should also avoid words that have more than one spelling if your visitors are likely to be confused and mistype the name. Alternately, you can buy both versions of the name and direct visitors from the one you like less to the preferred name. While you do want a short name (see below), don’t go for something so cryptic that people have a hard time remembering it. Word of mouth is a powerful marketing tool and you want to make it easy for people to tell their friends about your site.

In order to register any domain, you must provide certain personal information that you may not wish to be publically searchable in the Whois database. Signing up for WhoisGuard is a great way to keep your registration data private. WhoisGuard acts as a "shield" for your searchable information, displaying the address, phone number, and email of the domain registrar (Namecheap, e.g.) instead of your own. Keep in mind that WhoisGuard is an optional security add-on that must be renewed separately when you renew your domain.

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