A satellite navigation system does something conceptually simple and technically demanding: it works out where you are from how long signals took to arrive.
The basic idea
Each satellite continuously broadcasts its position and the exact time. Your receiver picks up the signal, compares the time in it with the current time, and calculates how long the signal travelled.
Multiply by the speed of light and you have your distance from that satellite. You are somewhere on a sphere around it.
Three satellites give three spheres that intersect at essentially one point.
Why four satellites
Because your receiver’s clock is not accurate enough. Getting the timing right to the required precision needs an atomic clock, which your phone does not have.
A fourth satellite provides an extra equation that lets the receiver solve for its own clock error as well as its position. That is why four is the practical minimum.
Why it needs relativity
The satellite clocks run at a slightly different rate from clocks on the ground – faster because gravity is weaker at altitude, slower because of their speed. Both effects are tiny and both are large enough that ignoring them would introduce errors of kilometres per day.
The corrections are built in. Satellite navigation is the most everyday demonstration of relativity there is.
Why it fails in cities
Signals reflect off buildings, so the receiver measures a path that bounced rather than a direct one, giving a longer distance and a wrong position.
This is why your position jumps around between tall buildings, and why phones combine satellite data with wifi networks and motion sensors.