The state of Ohio has a complicated system of local and city taxes. There are school district taxes, RITA taxes, CCA taxes, and some city taxes levied by certain cities. If you live OR work in those localities you are subject to the tax. Some of those returns can be prepared as a last step when you are in the state software program; for others, you are on your own. It is up to you to determine by doing a little "homework" of your own to see which local taxes you must take care of.
If your employer is withholding tax for the localities you work in---but do not live in--- you might not need to file returns for those unless they withheld too much and you need to seek a refund.
http://www.tax.ohio.gov/Individual/LocalTaxInformation.aspx
RITA Web site: http://www.ritaohio.com/
https://tax.ohio.gov/wps/portal/gov/tax/individual/school-district-income-tax
http://ccatax.ci.cleveland.oh.us/?p=ccamap
Q. Do I file with all of them, just one or none?
A. None
You normally only have to file a return in the city you live in. It is not necessary to file one in the city(s) you work in, if different. Your employer will remit withholding to the city you work in, but you do not file a return unless you're claiming a refund or your employer didn't withhold enough (both highly unusual).
Many Delaware County school districts do have a School District Income Tax (SDIT).
To find out if you live in a school district (and/or verify that you don't live in a city) with an income tax, enter your address at:
http://www.tax.ohio.gov/Individual/LocalTaxInformation.aspx