My family and I live in Pennsylvania. So my husband worked for a company whose headquarters are in New Jersey but the plants he did the actual work in are in New York State. I know we had to file PA state taxes but do we also need to file New Jersey and New York?

No need to file anything for NJ, as he neither lived nor worked in NJ (it doesn't matter that the company has its headquarters there). But he will have to file a NY State non-resident tax return since that is where he worked, and a PA tax return since that is where he lives. He completes the NY non-resident tax return first, which has him pay NY income taxes on his NY wages. Then he completes the PA return reporting all income from all sourrces (NY wages plus any income from other jobs, interest, dividends, etc), computes a preliminary tax on the total income, and then takes a credit for taxes paid to NYS. If his NY wages are the majority of income he may find that the PA tax bill is $0, or close to it.

I agree with ebaines with ONE proviso: Make SURE that your spouse's company does NOT withhold New Jersey income tax. The only taxes that should be withheld is New York, because that is the tax he will in fact pay. The credit noted in ebaines' answer should completely offset any Pennsylvania taxes owed.