Twitter outages are becoming much less common these days. In fact, there was a time not that long ago that the fail whale was an almost daily sight. Twitter is going back to its roots this evening with an authentication outage that's been dragging on for several hours already. So if you're unable to log in, don't worry, it's not a problem on your end.
Many users are reporting that they've been logged out of their accounts on Android and the web, then unable to log back in. No one's really sure what's going on, though the "down for maintenance" notice on the website is a little disingenuous. If anything, this is "unplanned" maintenance, AKA an outage.
Twitter's status page notes that some users are unable to sign in, but engineers are working to resolve the problem. There's a Tweet going around that claims to have found the issue—a coding error that is reporting a date in December 2015, which causes the authentication to fail. We're not completely sold on this hypothesis, but time will tell.
Update: It looks like service is being restored to most users now. It's still spotty on my end, but most of the AP staff can log in again.