Line 15 on W-2 shows NJ & NY. NJ has amount for line 16. NY has amounts for line 16 & 17. I worked in NY and live in NJ. What does this mean? Do I input both on taxes?

Yes, input your W-2 as is.

The way this works is this:

You enter your New York return (nonresident) first. You report your New York income in New York (TurboTax gets it from the W-2). New York tax is calculated.

Then you enter your New Jersey return (your home state). New Jersey is aware of your New York income. So, after the New Jersey tax is calculated, you are given a credit for any tax paid in another state (i.e., NY) on the same income. Look at line 42 on the NJ-1040. This should sharply reduce your NJ tax.

It's not perfect (you may still owe tax to one or both states), but it's how the states try to avoid double taxation.

The W2 shared by the employer is prepared assuming my work and residence location as NY. So, box 15 in W2 says 'NY' and box 20 mentions 'NEW YORK CIT'. 

How can I file the NJ resident taxes in this case ?