Part of choosing a good domain is getting accurate information, and knowing a few terms. Here's some information to get you started.
Mini-Glossary of Domain Terms
"Uniform Resource Locator" The address bar in your web browser where you see "http://www.mysite.com".
Top-level domain (for example in www.mysite.com: .com is the TLD, mysite is the second-level domain, and www is the third-level domain)
examples: .com, .gov, .edu, .org, .net, .info, .mobi
TLD with a country code
examples: .ca (Canada), au.(Australia), or .uk (United Kingdom)
Internet Network Information Center (aka Inter Network Information Center)
Internet governing and dispute resolution body in the U.S. Department of Commerce. InterNIC is a good source for a fast, effective search on domains.
If you're up for a little light reading about registrars, the domain name system and its history, you'll find some interesting intel if you travel more deeply into their website. InterNIC is licensed to ICANN.
Internet Corporation for Assigned Names and Numbers, operates the InterNIC web site.
Translated from the Geek, AJAX is a widely used web browser technology that delivers better, faster and more interactive web and web information applications. In short, with AJAX apps you can get more and better information faster, in a way that's easier to actually use.
A domain name registrar is an ICANN or national ccTLD accredited company who has the authority to register domain names. Registrars are competitive retailers of domain names. This means you can shop around for the best offers.
Best Registrars & Lookups
List of accredited registrars
Good lookup tool, also offers domain registrations
Good lookup tool
Domain selection tools allow you to search your desire domain name quickly and check the availability of your ideal domain name. They generally offer additional domain query support, and may or may not offer actual registration services.
Domainr finds domain names and short URLs and instantly check availability and register for all top-level domains (TLD's).
If you want only to check the availability of domain name quickly, try ajax domain selection tools. Domjax is a good choice, as the site makes it exceptionally easy to see if someone already thought of and registered that perfect domain name.
BustAName is a good domain name generator that's fun to use. It lets you enter words and word combinations. It will try every combination, then tell you which word combinations are available, and much more.
Impossibility is another good domain generator, also fun to use. It lets you enter parts of speech from a selection list, at either the beginning or end of your entered word.
Shop Around Before You Buy!
Remember that you don't have to buy through domain search services. You can find a name you like that's available, then buy from your registrar of choice, or through your web host.
Check to see who has the best price for 1, 2 and 3 + years. If you're buying a domain on sale, check to see how the regular annual renewal price compares to other registrars, and do the math.