I tried as one person suggested starting my Mac in "safe mode" this seemed initially to work, but upon re-starting my machine it was back to same problem with being "there is an error connecting to the Apple ID server"
This problem only happened after I upgraded from Yosemite to El Capitan, earlier in October(2015).
What worked for me was to use "Time Machine" and revert back to a previous time in September when I was using Yosemite. It did warn that you will loose things after this back-up date, but in practice once done, just about everything re-loaded from i cloud so was not a problem.
The following is from Apples instructions to revert to a previous OS X version
To do this restore, make sure your back up drive is plugged in then from the apple menu (top left) select restart. After your Mac restarts and you hear the startup chime (or just buzz) hold down the command and R keys, when the apple logo appears you can release the keys.
Select your time machine back-up disk (or external hard drive)
Select the Time Machine backup you want to restore. To restore your computer to the state it was in before you installed the new version of OS X , choose the most recent back-up that occurred before you installed the new version.
Follow the on screen instructions.
I agree this is very bad, and reminds me of Microsoft Windows problems, I will not upgrade again to El Capitan until these problems are fixed.