My daughter lives in New York, and worked for the summer in Massachusettes.
I can see from her W2's that no state taxes were taken.
Our accountant last year stated she had to claim for these un-taken taxes.
She paid approx 1-$200. She made approx $5K.

Some of her other friends who also worked at the same place and reside NY, rec'd money back on their taxes and stated they did not have to pay for those Mass taxes not taken.

I am confused. Please Help!

She will have to file a MA non-resident income tax return. It seems the other kids in the program had taxes withheld (I don't ubderstand why your daughter didn't) and received a full refund when they filed.

Not sure how they would qualify for a full refund on $5,000 in income.

Your daughter will need to file BOTH the Massachusetts and New York return, claiming the income on both. She can claim a tax credit for the Massachusetts taxes paid, so at least the income is NOT double-taxed.