Get your taxes done using TurboTax

Did you receive an actual 1099-MISC for this income?  Unless you did, you should report your fellowship in TurboTax Home and Business as follows: 

1. Go to the Personal Tab, then Personal Income, and say you'll choose what to work on. 
2. Scroll to the Less Common Income section and choose Miscellaneous Income (the last choice) 
3. Choose Other income not already reported on a Form W-2 or Form 1099.

1.  Answer Yes on the Other Wages Received screen.
2. Continue past Wages Earned as a Household Employee and Sick or Disability Pay.
3.  Answer Yes on the Any Other Earned Income screen.
4.  Choose Other on the Enter Source of Other Earned Income screen.
5.   Enter your fellowship information on the Any Other Earned Income screen
6. This will report your fellowship on line 7 of your Form 1040, and you will not be subject to Self-Employment Tax

As far as your states are concerned, it depends on your residency status. If you moved from one state to another and established a new residency, you'd file Part-Year Resident returns for both states, and pay taxes on the income that you earned while working in each state, plus any income from that state even when you weren't living there.

If one state remained your resident state, then that state would tax all of your income. The Nonresident state would also tax your income  that you earned there (there are two main ways that states calculate this), but your resident state would normally give you a credit for tax paid to another jurisdiction.