Double-taxed income is income that both states are taxing at the same time. New York is taxing you as a nonresident, and NJ as a resident, for the portion of the year that you were living in New Jersey.
To file this, first prepare a New York part-year return. The New York return will tax all of your income, since all was earned in New York. But New York will divide the income between your resident portion (January and February), and the nonresident portion (March through December) that is the double-taxed income. The New York return will ask how much of your income is the nonresident portion. In the box you can input the box 16 amount of your W-2 reported on the NJ line (line where box 15 says NJ). In addition, if you moved from NYC, also make sure that your move date from NYC has been inputted, because NYC does not tax a nonresident of the city.
After you have prepared the New York return, prepare the New Jersey part-year return. Because of reporting and programming requirements, the NJ return will add the amounts reported to each state together. At the end of this answer, you will see a key screen, and a box you must select to remove the double-reported income, so that the correct income is taxed. Then later, you are asked about the double-taxed income. If you have prepared the New York return first, the amount should be pre-calculated. If you don't see it, comment so that I can assist you to manually enter. It will take a little calculation because of your NY residency status.
Here is the key screen on the NJ return to remove the double-reported income:
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