I am filing taxes in NY state and on the W2 I received-there's a third copy apart from Federal and State tax filing copies that is meant for local filing.

It says at the bottom "Copy 2 to be filed with employee's City or Local Income Tax Return".

But as far as I know there isn't another tax return for NYC-so what do I do with this copy on W2?

Keep it for your records.

If I was expected to keep a copy, why does it say "filing copy" instead of "reference copy"?

They set it up for those who need the additional copy such as my area has a city income tax. If you are certain you don't need it, keep it or destroy it and just rely on the employee copy.

NewYork City also has a city income tax which has been withheld from me-I am not sure how to file for city income taxes though

See Personal Income Tax heading on the left side of this reference:
Finance - Forms & Reports (http://www.nyc.gov/html/dof/html/forms_reports/forms.shtml#personal)

I am not a resident of NYC nor am I am an employee.The link does not really tell me what to do with the W2 or what to do to get my check.


See Personal Income Tax heading on the left side of this reference:
Finance - Forms & Reports (http://www.nyc.gov/html/dof/html/forms_reports/forms.shtml#personal)

Actually I live in Brooklyn, NY (which I think might make me a resident of NYC for tax purposes) but in any case I don't find any separate forms to file for NYC income tax separately.

From that referenced site:
"Personal Income Tax
New York City charges a personal income tax to City residents however, the tax is administered and collected by the New York State Department of Taxation and Finance. "

Sounds like the refund would be factored in on the state form since the state administers NY City income tax, and the NY state tax form requests details about NYC residency on the first page. Follow the NY state instructions for NY City residents.

http://www.tax.ny.gov/pdf/current_forms/it/it201_fill_in.pdf

The NYC income tax is embedded in the NY state tax return.