I live in Ohio and work in Kentucky. My place of employment withdrew Kentucky state tax. Do I file Ohio state tax or Kentucky state tax or both?

You complete tax forms for both KY and OH. First do the KY non-resident tax calculation - they have form 740-NP which is used by both part-year residents and non-residents. Then do your Ohio tax calculation - OH first makes you calculate your tax as if you earned all your income in Ohio, then they give you a credit for taxes already paid to KY - see schedule D on page 4 of their IT-1040.