It seems like it would be a pretty straightforward task, but with so many new sites being launched each day, and so many individuals buying up available domain names to sell, this can be one of the most difficult decisions you'll have to make when it comes to launching a new website.
Fortunately there are some really helpful tools for finding and researching domain names. Take some time and go through the resources listed here and your search should be much less of a headache.
Whois will give you some basic information on the owner of a domain and the servers that are hosting it.
The WayBack Machine from Archive.org is another useful tool for researching domains. With this tool you can see how long a domain name has been around, if at all, which is something you're always going to want to know before purchasing.
Bust A Name
Bust A Name is probably my favorite tool for finding good, available domains. You'll enter a number of words that are relevant to your site (the more you list the more results you'll get) and it will show you any and all available combinations of those words. You can set it to create two or three word domains, and it will even give you synonyms for your suggestions.
Name Boy is a very helpful tool for finding available domains to purchase. Enter a primary word and a secondary word and it will produce a list of available results.
If you've hit a wall finding available domains, try Domain Hacks. It will give you some creative and unconventional ideas. Many of the will be domains you'd rather stay away from, but this is a great tool for seeing the domain finding process creatively.
Instant Domain Search
Instant Domain Search is a rather unique tool. It literally is instant. As you're typing a domain it's showing you available possibilities. Keep typing and you'll get some more samples listed. This can be a quick way to get some ideas flowing.
DomainsBot also works pretty quickly to give you a list of potential domains. It's not the best for finding new ideas, but it will quickly show you several domains along with information about their availability.
Just Dropped is a place where you can find domains that haven't been renewed by previous owners. You may be able to find some higher quality domains with this approach. Some of them will have a history with search engines, which can be good or bad, so do some research.
URL Trends is a helpful tool for researching existing domains. It will give you some information such as inbound links, PageRank and more. It's a good idea to use a tool like this before you buy a domain to see if it has any history.
Google Banned Checker
If you're buying an existing domain, you'll certainly want to know if it is banned by Google before purchasing. The Google Banned Checker is not going to be 100% accurate, but it might tip you off to a potential problem that you'd rather avoid.
Killer Domains E-Book
Daniel Scocco of Daily Blog Tips has written a detailed e-book on the subject of finding effective domain names. I personally own a copy of Killer Domains and I can say that it is a helpful guide if you're struggling to find the right domain.
Buying Premium Domains
In addition to the typical domain registrars like GoDaddy and Network Solutions, there are a huge number of marketplaces to buy premium domains for sale by the owner. Here is just a sampling of places to buy or sell.