Most non-nationals are liable for taxation on any income earned in the United States from the beginning of their arrival in the U.S.
Income can include salary, scholarships/fellowships, income from U.S. mutual funds or U.S. bank accounts.
Taxes for each calendar year are reported in the Spring of the following year and tax forms are filed with both the U.S. Government (federal) through the Internal Revenue Service (IRS) and the U.S. state(s) in which income was earned.
J Scholars and Professors are typically considered NR regarding taxation for their first two years in the United States.
H-1B, TN or O-1 status holders who have been in the U.S. for more than 183 days should go to the U.S. Internal Revenue Service (IRS) site resources to determine which form(s) they should file.
You should note that being a resident for tax purposes does not mean resident for immigration purposes.