Generally speaking, when a student is working for their school, they are exempt from social security and medicare withholding. The income is considered a student stipend rather than compensation for work performed.
Also generally speaking, you are required to file a state tax return in the state where your permanent residence is located, that reports and pays tax on all your world-wide income, even if you earn income while temporarily working in another state. Depending on the rules of the specific state, you may also be required to file a state tax return in the state where you are living temporarily, and pay income tax only on income earned in that state. Since your home state also taxes you on the same income, it will give you a credit to offset taxes paid to the state where your school is located.
Of course, if your income is less than the filing threshold, you may not be required to file at all. What state is your permanent residence located, where is your school, and what is your total income for the year?
