When you entered the U.S., you may have been provided with Form I-94 (Arrival/Departure Record) by U.S. Customs and Border Protection (CBP) at the port of entry. The Date of Last Entry is the date you last arrived in the U.S., and this is usually found in a stamp in the upper right hand corner of the I-94. If you do not have Form I-94, refer to your passport for the date of last entry into the U.S.
If you arrived to the United States via air or sea after May 2013, CBP probably did not issue you a paper I-94. Your I-94 Arrival/Departure Record is available electronically. Go to Where can I find my electronic I-94 record?
If you entered the U.S. without inspection (without papers), there is no official record of your entry. You will have to use a "Date of Last Entry" based on a best estimate.
In some cases, your port of entry to the U.S. could actually be in a foreign country. Therefore, your I-94 record (or passport) may have one date, but your physical arrival in the U.S. was a day later. In this case, use your port of entry date from the port of entry.
CitizenPath offers a quick and simple way to prepare your Form I-765, Application for Employment Authorization.