I second what Eigen said. You need to report your stipend net of qualified fees as income on your taxes, and will be taxed on a portion of it. Fellowship pay is reported in line 7 of the 1040 with "SCH" designated next to it. (See IRS Publication 970 Chapter 1.)
If you don't have the option of having taxes withheld from your paycheck, you should look into paying estimated tax. You may not have to in your first semester, but are likely to need to starting next spring if your fellowship and lack of withholding continue through all of 2016. The IRS has a great page with all the info you will need on estimated tax (mostly you will use Form 1040-ES). Basically, you have to project your income and withholding for the whole year and determine if you need to pay estimated tax, and then file quarterly if you do.
