According to the Google Adword Keyword Tool, tens of thousands of people search for the term “domain name” each month (just under 100,000 searches). Go Daddy is the largest domain registrar, with tens of millions of domain names under management, and NameCheap is on its way to 3 million domain names under its control (check out the company’s 3 millionth domain registration contest).
I use NameCheap (affiliate link) for all of my domain registrations. I migrated all of my old Go Daddy domains away from them last summer, before all the SOPA business had even started, and I’ve been pleasantly pleased. Their customer support has never been anything short of remarkable when I’ve spoken to them in the past. I highly recommend them.
I also hear a lot of good things about Hover. I haven’t used them yet, as I haven’t had need of a domain registration in quite some time, but I’ll probably register through them the next time I need a name just so I can form an objective opinion.
Either way, anything but Go Daddy seems to be the right way to go.