To ensure that we are all talking about the same thing when I say "domain name", let me briefly mention that a domain name is a name like "thesitewizard.com". Loosely speaking, it's the address of your website. Normally, if someone wants to set up a website, he/she will need to pay a small annual fee to register a domain name. The domain name can then be "pointed" to a website containing the web pages that the person wants to set up. Whenever visitors type that web address into their browser, they will be brought to the website. For example, if you were to type "thesitewizard.com" into your browser, you will be brought to the main page of this website.
Free Domain Name Registration
If you have a web site, you should seriously  consider registering your  own domain name. A domain name is a name like  "thesitewizard.com" or  "thefreecountry.com", which you can use to refer  to your website. Note  that you do  not have to be a company or organisation to register a  domain name. Any  individual can do it too.
Importance of a Domain  Name
There  are a number of good reasons for having a domain  name:
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     If you ever change your web host, your  domain name goes with you.  Your regular visitors or customers who knew  your site name as  www.thesitewizard.com (for example) would not have to  be informed about a  change of URL. They would simply type your domain  name and they'd be  brought to your new site.
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    If you  are a business, a  domain name gives you credibility. Few people would  be willing to do  business with a company with a dubious URL like   http://www.geocities.com/whatever/12345.
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    If you  get  a domain name that describes your company's business or name, people   can remember the name easily and can return to your site without having   to consult their documents. In fact, if you get a good name that   describes your product or service, you might even get people who were   trying their luck by typing "www.yourproductname.com" in their browser.
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    If you want good sponsors (advertisers) for your   website, a domain name is usually helpful. It tends to give your website   an aura of respectibility.
Registration Overview
Getting  a  domain name involves registering the name you want with an  organisation  called ICANN through a domain name registrar. For example,  if you  choose a name like "example.com", you will have to go to a  registrar,  pay a registration fee that costs around US$10 to US$35 for  that name.  That will give you the right to the name for a year, and you  will have  to renew it annually for (usually) the same amount per  annum.
Some  web hosts will register it and pay for the name for  free (usually only  the commercial web hosts), while others will do it  for you but you'll  have to foot the ICANN fees.
My personal  preference is to  register the name directly with a domain name  registrar rather than  through my web host. I've heard stories, in the  past, of  less-than-reputable web hosts that registered the domain under  their own  name, making them the owner of the domain rather than you  (although I  don't know if such web hosts still exist today).  Registering with a  domain name registrar allows me to make sure that I  am registered as the  owner, the administrative and technical contacts.  Being the owner is  vital — if someone else places himself as the owner  (such as your web  host), he can always decide to charge you some  exhorbitant fee for the  use of the name later, and there is little you  can do. The various other  contacts are less vital, but still play  important roles. For example,  the administrative contact's approval is  required before a domain name  is transferred out of a web host. If  he/she cannot be contacted, the  technical contact is used.
Although  some web hosts suggest that  you put them as the technical contact, you  may prefer to keep yourself  as one, so that when you want to transfer  your name to a new web host,  you don't have to wait for your old host  to approve the transfer.  Apparently a few have been known to take their  own sweet time to do  this, while unscrupulous ones have actually  refused to do it.
Domain  names disappear extremely fast. Many  people claim that all the good  domain names are gone. I doubt that —  but it is probably true that most  good domain names that are  descriptive of products and services have  been taken. If you want a  domain name for your site, I suggest you act  now, or face the anguish  of having lost that name later. After all,  US$10 (more or less) for a  year's ownership of the name is pretty cheap  when you realise that  you're cornering a good name for your website.
Step  By Step  Instructions
If you want to register a domain name,  here's what  you need to do. Please read it all before acting.
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     Think of a few good domain names that you'd like to use.  It won't do  to think of only one — it might already be taken (it  probably is!).  You can find some tips on choosing a good domain name  from my article,  Tips on Choosing a Domain Name, at   http://www.thesitewizard.com/archive/domainname.shtml
      There's more to it than meets the eye.
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    You will   need either a credit card or a PayPal account to pay for the domain.   This is a requirement of most if not all registrars. It will allow you   to claim and get the domain name immediately on application. This is not   an option (unfortunately).
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    If you already have a   web host, obtain from your web host the names of their primary and   secondary nameservers. Don't worry if you don't understand what these   things mean. Just save the information somewhere. The information can   usually be obtained from their FAQs or other documentation on their   site, usually under a category like "domain name" or "DNS" or "domain   name transfer" and the like. If you can't find it, email them. You'll   need the information to point your domain name to your website after you   buy your domain. Don't worry if you don't have a web host yet. Just   read on.
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    If you do not have a web host, you can   always allow the registrar to you to park your domain nameat 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, many registrars   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).
List of Domain Name   Registrars
There are numerous domain name registrars.   Listed below are just a few, along with my comments, if I know anything   about them. Note that the domain name industry is highly competitive,   with prices wildly fluctuating throughout the year, every year, so it's   impossible to really mention accurate prices below unless I spend all  my  time updating this page. Please check their sites for the latest  rates.  (Note: all prices are in US dollars.)
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     World's  Largest Registrar - GoDaddy.com — This extremely popular  registrar  (possibly the biggest today) offers .com domain names for  $9.99 (plus 20  cents) per year ($6.99 plus 20 cents if you transfer  from another  registrar). They have a web interface to manage your  domains, free web  redirection (where people who visit your domain will  get transferred to  another URL of your choice), free starter web page,  free parked page or  free "for sale" page, and an optional private  domain registration where  your domain is registered in the name of a  proxy company. They offer  .com, .us, .biz, .info, .net, .org, .ws,  .name, .tv, .co.uk, .me.uk and  .org.uk. Note that (as with all  registrars) the exact price varies  depending on which domain you are  registering (for example some domain  extensions are more expensive than  others). Both credit card and PayPal  payments are accepted.
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     Dotster.com — This fairly  popular registrar provides fairly cheap  domain prices ($15.75 plus 20  cents per domain), a convenient web  interface to manage your domains, an  optional privacy facility where  your domain name is registered in the  name of a proxy company, etc.  They offer .com, .net, .org, .biz, .info,  .us, .ca, .tv, .name, .cc,  .de, .sr, .md, .co.uk, .us.com domains, etc.  If you're transferring a  domain here from other registrars, the price is  even cheaper ($6.99  plus 20 cents). Both credit card and PayPal  payments are accepted here.
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    Register.com — This  domain name registrar has been in  business for a very long time: they  were one of the biggest around when  I started my first websites. They  are currently running an offer (only  via the above link) where they  charge $9.99 per year for a domain name  with a free business email  account. Although the offer will expire  eventually, you can apparently  continue to use the $9.99 price for the  life of the domain as long as  you stay with them. Domains qualifying  for this offer include .com,  .net, .org, .biz, .us and .info.  Country-specific domains have different  prices. Note: the special offer  link is rather flaky. If you don't get  the $9.99 offer the first time  you click it, but instead get their  normal $35 price, return to this  page again (for example, by clicking  the Back button on your browser),  and click the link again. The $9.99  offer should appear the second time  you click it (or at least it did for  me). [I've no idea why it works  that way; I don't run their site.]
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    Moniker— This  domain name registrar allows domain name  registrations for a plethora  of top level domains (TLD), including  .com, .org, .net, .info, .mobi,  .biz, .us, .co.uk, and so on (the full  list is too long to include  here). Prices for domain names differ,  depending on the extension.  Their normal prices are about $10.49 for  .com, $6.04 for .net, $10.95  for .org, $5.49 for .info, etc, although at  the time I write this  entry, it seems like they are having a sale, with  $7.59 for .com, $5.59  for .net, $7.97 for .org, and so on. Their web  interface allows you to  manage matters pertaining to your domain, such  as DNS, web forwarding  (where you forward visitors to your domain to  another URL of your  choice), etc. You also have the option to add "Whois  privacy", where  your domain is registered in the name of a proxy  company. Both credit  card and PayPal payments are accepted by this  registrar, although  PayPal payments have a surcharge.
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     1&1 Internet —  This is primarily a large web host that is also a  domain name  registrar. You are charged $6.99 for .com, .org, .net, .us  domains.  They also offer .info domains for $0.99 (first year only) and  .biz at  $8.99. The fee includes private domain registration, which means  that  your particulars are hidden from public view (done by registering  the  domain in the name of a proxy company). You also get a free email   account, DNS management, domain forwarding and masking, and a starter   website with each domain. Both credit card and PayPal payments are   accepted by this registrar.
In addition, as mentioned earlier, a   number of commercial web hosts will give you a free domain name if you   are Hosting with them, as will  many (if not all) of the registrars  above.
After You Register Your  Domain Name
If you're not sure  what to do after you register your  domain name, please read the The  Beginner's A-Z Guide to  Starting/Creating Your Own Website.