The issue seems to be infrastructure-related, with users encountering "server time out" errors after entering their credentials. IsItDownRightNow shows that the service is currently suffering an outage. Apple's own support system is also listing that iCloud-based services — including iCloud Account & Sign In, iCloud Backup, iCloud Drive, iCloud Keychain, iCloud Mail, iMovie Theater, Photos and iWork for iCloud beta — may be slow for some users.

Update: Apple is now saying that the iCloud issues have been resolved. Overall, the company says that about 40% of all iCloud users were affected by the disruption.

Source: Twitter, Reddit, Apple