The employer is using a table based on your W-4 information that you filed with him/her to do a look-up in a table to guesstimate a proper amount of withholding.
On the W-4, you may have indicated many more exemptions than you were due, in order to increase your take home pay.
If your income paid by the employer was not very large, then it might have been appropriate to withhold nothing - your employer doesn't know your full tax picture (married? kids? other income?) and can do the table look-up based only on what he/she is paying you.
Now that you see the full picture, you can work with your employer to adjust your state tax withholding to approximately cover the expected balance due for next year.
On my W4, I claimed 0 for both federal and state. Federal income tax was withheld, but not state. How do I change my W4 to have state tax withheld if I already entered 0?
You can't change it for 2017. You can change the ongoing W-4 for 2018 by going to your employer and asking what you have to do to get some amount of tax withheld for the state.
It is a bit odd that 0 exemptions produced no withholding, but it may be that you are in a state that has a high income threshold for taxation. It really varies from state to state - speak to your payroll department.
I'm in California. I found a de4 form - maybe I need to give that to my employer for this year. My company said I need to talk to ADP. I called ADP but they don't talk to employees. So the circus continues!
Does it make sense that if I don't make enough income to pay State Tax, I still have to pay State Tax on my 2017 taxes? That doesn't make sense to me.