Here is another similar "GPS for dummies" link just like the one above..
GPS and relativity
A good popular article, as I mentioned it before, is from renowned Prof. Neil Ashby in Physics Today. Here is a link I think will work without subscription.
RELATIVITY AND THE GLOBAL POSITIONING SYSTEM
I used to visit U of Colorado, boulder in 70's in summer - (Nice place to spend summer and visit Rockies). Prof Ashby was even then a big shot there. Boulder also had George Gamow on its faculty at one time in the past. George Gamow's "One Two Three Infinity" and "Mr. Tompkins in Wonderland" are two of my most favorite popular books to learn theory of relativity. (Shivji - may remember it, I recommend it long time ago)
Generally we need Einstein's general relativity to understand extreme astrophysical realms.. blackholes and such.. and until recently (before 1975) there was no easy way to confirm it with experiments on earth. In practical every day life GPS is the only application I can think of where the theory turns out to be essential for engineers to worry about.
(In fact I still remember the suspense when first atomic clock went up and many were skeptic that it will confirm Einstein's theory of gravitational time-dilation .. turns out that it did, and it was a big news then)
Unfortunately math needed to understand general theory of relativity is hard, and many results are quite counter intuitive.
The above article, IMO, is quite comprehensive and may able to answer some of the questions raised by JayS, Vina..