Most modern computer programs store the date using a 32-bit integer counting the number of seconds from midnight, January 1st, 1970. This integer has a range from -2,147,483,647 to 2,147,483,647. Right now that number is around 1,083,006,000.
So when does this system stop working? On Tuesday, January 19th, 2038 at 3:14:07 AM (GMT), the clock will reset to Friday, December 13th, 1901 at 8:45:52 PM. At that moment, airplanes and satellites will realize they have not been invented yet and will fall right out of the sky.
We have less than 34 years to get ready for the end of civilization as we know it. You have been warned!