The distance from each satellite to your position is calculated from the time it takes for the signal to travel the 20,200 km (12,600 miles) when overhead, which goes up to 26,600 km (16,500 miles) when on the horizon, to your receiver. With the signal travelling at 300,000 km/s, the time taken is between 89 and 67 milliseconds, and so this has to be measured with nanosecond accuracy. One of the most remarkable things about GPS is how the cheap and simple clock in the receiver is made to have the same accuracy as the very complex and expensive clocks in the satellites.
Darron explained how the fix is obtained using four satellites. One defines a sphere, the second intersects this as a circle, the third cuts the circle in two points and the fourth distinguishes these two points. If the assumption is made that the receiver is near the surface of the Earth, then this can be used instead of the fourth satellite measurement. Ideally these should all intersect at a single point, but in practice, without correction, they would be spread out slightly due to the receiver clock running fast or slow. By adjusting the clock rate to get as close a match as possible, the timing accuracy needed is obtained. The remaining spread is a measure of the accuracy of the fix.
In the old days of navigation by chronometer for longitude, all that was really necessary was an extremely stable clock. Although the clock would run slightly slow or fast, this did not matter as long as the rate was known; it was easy then to calculate the exact time from the rate and how long since the clock was checked against an accurate time, such as the firing of a noon gun in port. Similarly what is really needed in the GPS receiver is a simple but stable clock, with the rate worked out as above, to give you the equivalent of an "atomic clock in your hand".