GPS is probably the most familiar real world application of Einstein’s Special Relativity. The GPS system relies on nanosecond (billionths of a second) accuracy of time codes to calculate distances between the user and the satellites. Due to the high speed at which the satellites much orbit, special relativity tells us that relative to our view point their clocks roll about 7 microseconds (millionths) slower per day. On the other hand, since they are further away from the gravity well of earth they experience a relative time that seems 45 microseconds FASTER! What this means is that if we didn’t have a deep under standing of Special Relativity they would be off by 38 microseconds (45-7) each day. While 38 microseconds may not seem like a lot, this would make the GPS nearly useless after only 2 minutes. By the end of the day, your GPS location would be off by 10km(6.25mi)!
Learn more at: http://www.gps.gov
Special Relativity explained quick and simple: Minute Physics (also cats)
More in depth, but non-mathy, relativity videos: PBS Space Time
Math reference: http://www.astronomy.ohio-state.edu/~pogge/Ast162/Unit5/gps.html