If you're a visitor to the United States, but your departure flight has been canceled due to the ash from the volcano in Iceland, you are probably facing a host of inconveniences. Perhaps you're figuring out where to stay, and talking things out with your employer back at home.
Unfortunately, you may have one more inconvenience to deal with if your permitted stay in the United States is about to end. The fact that you're trapped here does not, unfortunately, give you any automatic right to stay beyond the time permitted according to your visa or, if you traveled without a visa, by the Visa Waiver Program (VWP)
The first thing to do is figure out the exact time limit on your stay. If you traveled here on a visa, DON'T look at the date on the visa itself -- that only tells you the last date on which you could have used that document to enter the United States. Look instead at the little white card called an "I-20" that you were given by the airport or border immigration authorities when you entered the United States. It will have the date by which you are expected to leave.
If you entered on the Visa Waiver Program, you would have been given a little green card with your expected departure date. The maximum you could have been allowed in for is 90 days.
Next, consider what consequences you might face as a result of an overstay. For most people, the biggest problem is that the original visa will be voided (canceled). That would mean you'd have to get a new one before returning to the United States -- which will forever be harder with an overstay on your record.
A few people, however, face even worse consequences. If their overstay ends up lasting 80 days or more (but less than one year), and they leave the U.S. without removal proceedings having been started against them, they are barred from reentering the U.S. for three years from their date of departure. (Those who overstayed for more than one year face a ten-year bar on returning.)
So, what do you do to avoid these consequences?
If you're on the Visa Waiver Program, USCIS expects you to either visit its local office or, if you happen to be at an airport, visit U.S.Customs and Border Protection at its office there. To visit USCIS, you must make an appointment first, using its online INFOPASS system.
If you're traveling on a visa, extending your stay requires filing a request with USCIS on Form I-539, Application to Extend/Change Nonimmigrant Status. You must do this before your authorized stay expires. You'll find the form and instructions on the USCIS website.