Can't log in to FaceTime
There are a couple of things to try.


In System Preferences > iCloud, log out and then log back in.
If that doesn't work, contact Apple Support, per this post in the Apple Discussion Forums:


  You need to call Apple. The problem is NOT on your end! I spoke with an Apple representative and they said that it is a "registration issue" and that THEY need to reset security settings. I waited 14 minutes but the representative found the issue then reset it and I was able to sign in.
  
  Call 1-800-275-2273 (It's free in the US or on Skype if you're in Europe!)  When you call, you'll get the standard automated secretary.  DO NOT SAY ANYTHING.  If you say anything, you will be told that you either need to purchase a protection plan for $19.99 or literally "end the call". Just stay on the line until you're asked if you wish to speak with a representative. Do that and you'll be connected (for free).  They will ask for you Macbook Pro serial number and a "Customer Code" if you have one.  Tell them how long that you've had this issue and they'll confirm your Apple ID.
  
  Once they confirm your Apple ID, they reset everything on their end and you should be set. Make sure that you are able to send messages/make calls before hanging up.