The income tax system is pay as you go. If you don't pay tax into the system through withholding, then you are responsible for making your own tax payments, usually quarterly. (The federal payments are not 4 exact quarters; you pay estimated taxes April 15 for income earned January-March; June 15 for April-May income; September 15 for July-August income; and January 15 for September-December income.)
If you end up over-estimating and paying more than you owe when your tax return is calculated, you get the difference as a refund.
Estimated payments are also required when you have a lump sum of income, such as selling an investment, where the taxes won't be covered by your regular job withholding.
In your case, since you receive a fellowship and the university does not treat you as an employee, you are responsible for making those estimated tax payments. Payments can be made by check or electronically. See more here.
https://www.irs.gov/businesses/small-businesses-self-employed/estimated-taxes
https://www.michigan.gov/taxes/0,4676,7-238-73294_73296_73297_75057_77806---,00.html
When you don't make the proper estimated payments, you are subject to a late payment penalty, even if you pay in full when you file your tax return.
For MI-2210, your quarterly earnings is simply your monthly stipend x the number of months in the quarter.
If you also had a side hustle (Uber, Doordash, etc) these earnings are also subject to estimated taxes and would be included in your quarterly earning to calculate the estimated amount of tax owed and now the estimated amount of penalty.
Note you are also required to make estimated payments for the 2020 tax year, although the April 15 and June deadlines for the 1st and 2nd quarters have both been extended to July 15.
