The formula
If you aren't a US citizen or green card holder, you meet the test for a calendar year when you were physically present in the United States on at least 31 days that year, and on at least 183 days over three years counting all the days this year, one third of the days last year and one sixth of the days the year before. The rule is in Internal Revenue Code section 7701(b), and IRS Publication 519 explains it.
Example
120 days in each of three years: 120 + 120/3 + 120/6 = 120 + 40 + 20 = 180. That's under 183, so the test isn't met. Three more days this year would reach 183. That's why many Canadian snowbirds hold to about 120 days a year.
What counts as a day
Any part of a day in the US generally counts. Some days don't, including days in transit between two foreign places for less than 24 hours, days you commute from Canada or Mexico, days you couldn't leave because of a medical condition that arose in the US, and days as an exempt individual such as certain students, teachers, trainees and diplomats. US territories aren't part of the United States for this test.
The closer connection exception
If you meet the test but were in the US fewer than 183 days this year, kept a tax home in another country all year and had a closer connection to it, you may be treated as a nonresident by filing IRS Form 8840 on time. A tax treaty can also change the result. Ask a tax professional about your situation.
Track it automatically
The day tracker counts your US days from your log for this year and the two before, and shows how many days are left before you'd meet the test.