This is as crazy and dynamic a question as any. The quick answer is somewhere around 240 Million domain names have been registered, but there are only 80 million of those (or roughly 1/3) of those are active.
What’s the difference between registered and active domains? Registered means someone owns the name, and active means someone has actually made a website for that domain.
This information was taken from netcraft.com and is the most accurate count as of May 2010.
Here is the link to the actual survey if you want to read more.