"Rollover" (when the game resets the time of day and adds more turns) occurs at roughly 11:10 p.m. (just before midnight) Eastern Standard time. It only takes a moment, and if you're logged in when rollover occurs it will log you out while it's updating your information.
Currently in Maintenance
Hiya! You've caught the game while it's in the middle of its nightly maintenance, which we normally call "rollover." Rollover usually begins at roughly 11:10 P.M. Eastern Standard Time (GMT -5) and runs for just a few minutes. The game ought to be back online shortly. For anyone bad at maths the GMT time for rollover is 4:10 am. *Yawn*.
In the meantime, if you'd like something to do, you might consider visiting the Twilight Forums.
Does the cat in the picture have a name?