You file a full year resident MA return and pay MA tax on all your income regardless of where earned. You file a non resident return for CT and pay CT tax on the income your earned (from your out of state employer) while living there. MA will give you a credit or partial credit, for the tax you pay CT.
If your employer is in NY, you may be subject to withholding for that state too. If you work outside NY as a job requirement, you are only subject to New York State income tax on the days you work in New York. But if you work outside New York for your own convenience, you are subject to New York State income tax on all your income. Pennsylvania, Nebraska, Delaware and Arkansas have the same rule.
