State tax filing

This is correct.  This is a reporting requirement for your employer to report for your W-2 income.  You will want to do the following:

1. Prepare the New York nonresident return first.  Both states tax your income, but NJ will give you a credit for the tax you must pay to NY on the income earned there.  By preparing NY first, it is easier for TurboTax to calculate the credit.
2. On the NJ resident return, you will be asked if both of the lines on your W-2 (NY and NJ) are taxable in NJ.  They are not.  Only the NJ income is, and you will mark the NY income as not taxable in NJ.
   
3. A few screens later, you will be asked about double-taxed income.  This is to calculate the credit in point 1.  You will enter the NY amount again, which is being taxed in both states.  The slightly higher portion to NJ is taxed only in NJ.  New Jersey does include some items as taxable income that are not taxable to either NY nor the IRS, as this website shows:  NJ Division of Taxation - NJ Income Tax Wages - State of New Jersey

Here is an FAQ to assist you to prepare the nonresident return:  https://ttlc.intuit.com/replies/3302052