I live in PA and work for a corporation in NY State. I have PA state taxes taken out of my check, but not NY states taxes. When I file at tax time, will I have to also pay NY state tax? Should I have both PA and NY states taxes taken out of me during this year or what?

If you are living and working in PA, you do not owe NY state tax. You do not have NY income. All your worldwide income is PA income.

If you are present in NY to work, then you have NY income. Your U.S. Tax Return: Working or Living in Two or More states

Since you commute to work in NY, you should have NY taxes withheld from your pay, as your income earmed in NY State is taxable there. You will have to file a non-resident NY State tax return, as well as a resident PA tax return. On the PA return you report all income, but take a credit for income tax paid to other jurisdictions (NY State in this case), and that credit will make it so you don't owe PA any tax on the income you earn in NY (since NY State's tax rates are generally higher than PA's).