creative cloud does not use a serial number and creative cloud subscribers to not have any serial numbers associated with their programs. creative cloud subscribers use their adobe id to activate their programs.
download and install the creative cloud desktop app (after signing in with your adobe id - the same one you used to pay for your creative cloud subscription), https://creative.adobe.com/products/creative-cloud
open the desktop app (a shortcut will be on your desktop) > click apps and then select the programs you want to install.
the desktop app gives you acces to cs6, cc, cc 2014 and cc 2015 apps. and probably more in the future.
if you already have the cc desktop app and you need to refresh your adobe id, sign out and then back in to your cc desktop application using the same adobe id used to pay for your subscription: http://helpx.adobe.com/creative-cloud/kb/sign-in-out-creative-cloud-desktop-app.html
Thanks for the response, but this guidance does not really speak to my particular situation. I'm not an individual purchaser, my organization purchases bulk licensing so the Adobe ID I log in with is not what was used to purchase the license. I was given a registration key to use from my organization, which was provided by Adobe. I already have the CC desktop app and there is no way to tie the two together. I guess I will just have to wait until regular business hours to get a straight answer from Adobe. This new licensing model is clearly not really friendly to large organizations.
I'm in the same situation... was given a serial for PS by our admin and told to download the app at Adobe. So... how does this work?