This is an old question, but this happened to me today as well. I resolved it.
iCloud was asking for an app-specific password. This is different than your Apple password, or your computer/iPad password. To generate an app-specific password, go here:
https://appleid.apple.com/account/manage
and scroll down to "Security," then click the little "Edit" link over on the right. (This information is BURIED on Apple's web pages; I found it in my browser history, not via any sensible navigation. Shame, Apple.)
Scroll down to "App-specific passwords," click "Generate password," name it something (e.g. 'iCloud'), and then copy and paste that password into the dialog box that keeps appearing. I had to paste it in twice, and get two text messages with login codes, but it has (for this morning!) solved the problem.