Hosted solution question - Detecting all registered domains

I recently released a turnkey script (like WordPress) and am now looking to release a hosted solution.

I would like customers to register the domain name themselves. Then, when they sign up w/ my hosted solution, they specify their domain name and point their nameservers to mine.

Then, my service will automatically set up the site under their domain name. I would like all customer domains to point to the same set of script files, so that I will only have to push updates to a single install of the script and it will be reflected across all customer sites/domains.

The script will detect the customer based on the domain and deliver the relevant content.

So my question is…

How do I set up my script or server to detect the customer domains? I assume it’s some DNS setting?

Can anyone shed some light on this? If you could provide tutorial links related to this, that would be awesome.