Through the course of any given week I find myself checking for domain name availability ten or more times. I've used each of the following domain tools at one point or another, and they each have a different approach.
Moniker is usually my first choice for domain name research. While the interface is basic, it allows for up to 500 domain names to be checked at the same time. The tool will automatically remove invalid characters as well, which is great if you're used to using exact mode for Google Keyword Research. You can paste a list directly from Google Keyword Research -- including the brackets -- and it will remove the brackets and check the domain availability for all those terms.
While it would be better if registration wasn't required, the ability to check so many names at once more than makes up for the inconvenience.
Domain Typer is similar to other tools on this list in that it uses Ajax to quickly determine a domain's availability. Additionally, it will attempt to create a "domain hack", which is a creative way of saying they try as many country code variations as they can to try to match your entered term. Thus, while monkeyfood.com is taken, perhaps you'll consider monkey.fo/od, using the .fo country code of the Faroe Islands.
Similar to Domain Typer, Domai.nr excels at quickly using domain hacking to try to find available variants of the terms you enter.
Name Boy does a great job of exploring synonyms of the keywords you enter, and then checking the availability of the domain name combinations that result. Name Boy is the only tool on this list that will also check rhyming words as well.
By entering several keywords, Bust a Name will try each combination of those terms in succession and show you which of those are available. Usually by the fourth keyword you will have created a list of available domain names that you likely had not yet considered.
Who is My Project lets you do quick research to find out how popular your domain is in Google, and offers a drag n' drop system for managing the domains you like.
Undoubtedly the most creative approach to domain research on this list, Wordoid comes up with fictional words that "sound like English". This may sound unusual but it's an effective way to brainstorm domains you may not normally consider, especially for Web 2.0 properties.
A quick an easy way to review domain databases, Domains Bot does also does a great job of showing you variations on your terms that you may not have considered. It will include hyphens in-between words, as well as add interesting adjectives before your domain request. Did you know that EvilMonkeyFood.com is available? Hmmmm...
Domize takes an interesting approach by quickly showing you the availability of all the shortened versions of the domain you entered. So in addition to searching for monkeyfood.com, they will search for "monkeyfoo", "monkeyfo", "monkeyf", and so on.
You may scoff at the idea, all I know is that I didn't know I wanted monkeyfoo.com until I saw it on Domize.
Domain Tools does an excellent job of checking the most common domain databases to see if your chosen name is for sale, or has been deleted from the domain registry recently.
The biggest draw for Ajax WhoIs is its speed. Checking the availablity of a domain is quick and easy, yielding fast results.