Cannot Sign In To Mac Messages Or FaceTime

I think most Mac people use the Messages App for their texting needs. There are other third party Apps available, but Messages is more than adequate for most people. Despite its faults, Messages is a pretty reliable App. However, on occasion, some people are unable to login to Messages and/or FaceTime (apparently this problem affects both Apps). The causes of this problem are unknown. If you have had or are having problems logging into Messages or FaceTime, then read this article listing the ever more complicated troubleshooting steps.