With the disclaimer that I am not a tax expert and am not offering financial advice, here's essentially how it works:
Federal taxes aren't withheld from stipends funded by NIH training grants, since these aren't considered "salaries" per se (according to the NIH). However, students will owe taxes on this income in April (the IRS trumps the NIH). Often, students who are US citizens / permanent residents receive the majority of their funding from these sources during their first year or two in the program.
Funds from non-training grant sources are often used to supplement the amount provided by the training grants, and these are taxed each pay period. For example, you may receive $23,000 of your salary from a training grant -- federal taxes aren't withheld from this amount. They will, however, be withheld from the remaining amount. Clear as mud. Stipend amounts are listed on form 1098 which you can access via MyUW.
International students and U.S. citizens / permanent residents who are not funded by training grants are generally funded via research assistant appointments either via the department or their thesis labs. Federal taxes are withheld from these salaries, so while gross pay is the same, the net salary will be less. The benefit of sorts is that you won't have to come up with all this cash at tax time.
The simplified version: Students funded by RA appointments pay taxes now; students funded by training grants pay taxes in April (or, much more often, make quarterly payments).
For more help: The resident tax expert in Student Fiscal Services is Lichang Wong - lichang [ a t ] u.washington.edu. Student Fiscal Services holds tax workshops for graduate students several times each Spring - if your funding source includes a training grant, please consider making time to attend one of these to save yourself some frustration in April.
IRS publication 970 spells out the rules. Their chart indicates you must pay taxes on your stipends, but not for funds spent on textbooks or other supplies.
More tax info is available on the UW website.