At the time of writing, Xbox LIVE is currently not allowing users to sign in. This problem has persisted for an hour and a half so far, and Microsoft are currently "working to resolve the issue." Whatever that may be.
Keep calm and carry on. You can check the status of the service here.
UPDATE: Signing in now seems to be working for some users - at the very least, I've been able to load a few adverts. Many gamers are reporting normal service, so with luck, things should be sorted very soon.