During my childhood I've been to the US many times. I always realizes that when I entered the US my passport was stamped with the date of my entry and when I returned to my homeland (Jamaica), they would stamp it to show when I returned. Last summer (2016) I visited the US and as usual my passport was stamped with the date of my arrival. When I got back home to Jamaica I went through immigration and all. When I left the airport I realized that my passport wasn't stamped with my return date. Is this a serious problem or the US stores international departures record. I'm confused and I need help.
The US keeps their own records, so this should not be a problem. Many countries do not stamp passports for their citizens (or at all), so this is quite typical.
As phoog suggests, you can check your record by going to i94.cbp.dhs.gov and filling out the form. If they have your departure recorded accurately, which will normally be the case, then you're all set.
If that webpage shows that the US Government does not think you left on time (glitches can happen, especially if you enter by air and depart by the land borders), they advise:
CBP tracks departures in a variety of manners, but not all departures are available on the website. If you are departing by air or sea, your departure is recorded electronically. There may be times when this information does not appear accurate. There is no formal action to take. CBP will not update the records available on the website.
Also, some land border departures may not be recorded in CBP systems. Land border departures on the Southern Border are not always documented. A departure will be recorded if you depart via land and re-enter the United States prior to the expiration date stamped in your passport. If you are not a resident of Canada or Mexico and you receive an electronic I-94 and depart via land, but do not re-enter the United States prior to the expiration date stamped on your passport, you may want to travel with evidence of your departure into Canada or Mexico. Evidence of departure can include, but is not limited to, foreign entry stamps in a passport, transportation tickets, pay stubs and/or other receipts.
If this is the case, it's a good idea for the next time you come to the US to carry some proof that you returned to Jamaica on time, such as your boarding pass stub, in case you are asked about it.