Of all the processes involved in setting up and managing a website, the domain name registration is one of the most important and one that you should get right first time. If you've chosen a good domain name registration agent/agency (also known as a registrar) but chosen a poor web host then you can easily configure your domain name to use a different host and solve the problem. But, if you have a poor registrar then you may find yourself not only stuck with a poor host (possibly the same company as your registrar) but you may also find that you're not the legal registrant and owner of the website domain name (such is often the case when an agency uses an agency of its own, thus the agency that you've chosen is the official registrant and their agency is the official registrar).
USA owns the Internet and its World Wide Web, and the USA have put their organisation called ICANN in charge of delegating ownership of websites to the national authorities of each country, which in turn delegate domain names to the general public.
ICANN is the International Consortium of Assigned Names and Numbers. This means that they control who owns each domain name and which web servers they point to (referred to by nameserver such as ns1.bwdp.org or by numerical IP address for example 123.123.123.123).
In the important decision of choosing a domain name registrar, it's strongly recommended to choose from this long list of ICANN accredited registrars. Not only does this guarantee that your registrar has to some extent been vetted by ICANN and approved, but it ensures that the registrar conforms to ICANN's domains dispute policies. This means many things including that any disputes over ownership of a domain should be settled under the jurisdiction of the country corresponding to the domain name extension (for example a .org domain name dispute is settled in USA and a .co.uk dispute is settled in the UK).
When registering a domain name it's worth thinking about how memorable the domain name is as well as what percentage of potential users are likely to mistype your domain name. When accounting for spelling mistakes and typographical errors, if you're planning to register a .net domain it may be worth registering the .org (if available) also; and if registering .org.uk then consider registering .org also (if available). And if you have any double letters, such as in "Google" and "Yahoo", then consider registering the domain name corresponding to the double letter being typed as a single letter (for example "Gogle" and "Yaho").
When choosing a domain name for your website, consider which buzzwords you're primarily optimising for and choose a name that contains these words. This will ensure that a large proportion of the people who link to your site use the best keywords in their anchor text. This assists in increasing your rankings in search engine results pages (SERPs) when people are searching for phrases that match your keywords.
If you want a short and snappy domain name which would be easy to remember and easy to type, consider choosing an acronym standing for a larger name which contains your keywords when written in full. For example, BWDP stands for Birmingham Web Development Partnership; thus bwdp.org.uk is a well chosen domain name when optimising for the popular keyword "Birmingham" or the popular phrase "Web Development".
It might be worth adding miscellaneous non-keywords for the purpose of branding. For example, gocompare.com is not as keyword-rich as compare.com, but at the time of writing this article, gocompare.com has a greater brand identity and is about 5 times more popular according to both Google PageRank and Alexa Traffic Rank.
When you register a domain name you'll usually need to give your personal contact details (such as name, telephone number, email and postal address) to your registrar. You'll also need to tell your registrar which web server (nameserver or numerical IP address) you want the domain name to point to. The registrar will then pass these details to ICANN (usually instantly or within a few hours). ICANN then stores a publically-viewable database record about your website which includes information such as date registered, date of expiry, applicable web servers, and your personal contact details.
To prevent ICANN from publishing your personal contact details in their public database, which is accessible via various Whois lookup tools, ensure that your registrar either offers a "proxy" registration service to protect your personal information, or use some friend or affiliate organisation's contact details (but be sure to use at least your own real initials in order to be the legal registrant and official owner of your domain name). If you don't mind dealing with the potential of being harrassed though, then using your proper contact details is the best way to avoid domain name ownership disputes.