Following the launch of iOS 8.1.1 and OS X Yosemite 10.10.1, it appears that Apple's servers are having some issues with iMessage. Multiple users on Twitter and other social networking platforms are reporting that iMessage is down, and MacRumors has confirmed that some messages appear to be hanging for quite some time before sending, while others are simply not delivered.
Currently, Apple's System Status page is not reporting any iMessage outages, and some users appear to be experiencing no issues at all.
The iMessage issue is not a result of the iOS 8.1.1 update, as users who have yet to update are also reporting problems. The outage is likely to be temporary, with service restored shortly.
Update: Apple has documented the outage, which affected "some" iMessage users, on its System Status page. The outage lasted roughly 90 minutes and was fixed by 11:30 AM Pacific Time.