Update: As of 3:02 Pacific, the iCloud login issue has been resolved.
iCloud System Status (timeline shown in HST). | Source: Apple
According to Apple's System Status webpage, the iCloud issues began a little before 2 p.m. Pacific, a few minutes before the company released the latest Mountain Lion update.
The problem, which manifests itself as a password or user ID error, is limited to iCloud's Account and Sign In feature, which could affect user access to other services like Messages. Apple notes that all other iCloud assets operating normally, meaning that users who are able to authenticate their accounts with the attached services will go unaffected.
From the Status Webpage:
Users may incorrectly receive an authentication error when attempting to use iCloud. Setting up a new iCloud account from an eligible device may be temporarily unavailable.
It is unclear if the iCloud problems are related to the OS X release, and Apple has yet to offer additional information regarding the matter.