Once you have fallen out of status—meaning that your official authorization to stay in the United States on a visa or some other document has expired—you are expected to depart the United States immediately, or within the relevant grace period. Unfortunately, at this point you're not likely to be eligible for any sort of work permit, or indeed for any other immigration benefit. Your current visa will be automatically voided (under § 222 of the Immigration and Nationality Act (I.N.A.)).
And there's more bad news. Having fallen out of status, you are also at risk of being denied future visas to travel to the United States, or being barred from reentry to the United States for a certain period of time, as a penalty for unlawful presence during your overstay.
If you last entered the United States illegally, without a visa, then you never had any status in the first place, so most of the discussion in this article will not apply to you. But an illegal entry causes various other difficulties in seeking a U.S. visa or green card, as you are immediately deportable, and will also face penalties for your unlawful presence in the United States.
Before worrying that you must leave the U.S. right away, make sure you understand the date by which you are expected to depart. That date should be shown on your I-94 Arrival-Departure Record, which could be either a small white or green card that was tucked into your passport when you entered the United States or simply a digital document, which can be obtained from the website of Customs and Border Protection (CBP).
Notice that that the I-94 shows a different date than was on your original visa. A visa expiration date refers to the last date on which you can use the visa to enter the United States, but tells you nothing about how long you can stay in the United States once you're there. (To learn more, see I-94 Date vs. U.S. Visa Expiration Date.)
If you entered the United States as a student, your visa might say "D/S" instead of a date. That indicates "duration of status," which means that you may stay in the United States for as long as it takes to complete your studies (up to the projected end date shown on the I-20 form you received from your school).
In many U.S. visa categories, you can renew or extend your status before it expires by submitting an application to USCIS. The form that's usually used for this is called an I-539.
If renewals are allowed in your visa category, make sure to apply well ahead of time, so that you don't fall out of status. Once you reach the expiration date on your I-94, or are otherwise out of status, you can't file for an extension.
If your authorized U.S. stay has ended and you are simply wishing you could stay around longer and find a job, you are probably out of luck.
For one thing, before applying for a U.S. work permit, you would need some sort of underlying basis of eligibility, such as a job offer from a U.S. employer that is willing to help you get a temporary visa or green card, or immediate eligibility for a green card through a family member. But even if you were eligible for some sort of visa or green card, having stayed in the United States illegally could make it difficult or impossible for you to successfully complete the application process.
Nevertheless, many exceptions exist within the world of immigration law. If, for example, you fear persecution in your home country, applying for asylum could eventually lead to a work permit and/or permanent status in the United States. And if you were to marry a U.S. citizen, and your last entry to the U.S. was lawful (not, for example, an illegal border crossing), you might be able to apply to adjust status and get a green card on that basis.
So, if you believe that you are eligible for some sort of temporary visa or permanent green card, speak to an experienced immigration attorney.