Whether you have a regular job or a fellowship (like the NSF Graduate Research Fellowship, or a Postdoctoral Fellowship), you need to pay taxes. And really the one (very small) downside to having a fellowship is figuring out and paying your estimated taxes. Lots of other people (self-employed, for example) also have to figure out their own taxes. And, if you owe more than 1,000.00 for the year, the government requires that you pay estimated taxes (in no longer than quarterly intervals). It can seem a little overwhelming. Luckily, we live in the electronic era and there are two freely available tools (from the government, no less) to help you.
First, you can use the withholdings calculator provided by the IRS to get a handle on what you will likely owe. Way to go IRS, for making something user-friendly and extremely useful! Then, once you have estimated out how much taxes you will owe, you can pay them online using the Electronic Federal Tax Payment System (EFTPS). I am very glad to live in a time where I have these electronic tools to help me, instead of having to figure them out by hand, and mail in physical checks, wondering if they got lost in the mail, or are postmarked on time.