That imaginary line is a relatively modern invention. Nobody really cared about longitude until it became the key to successful ocean-going trade. Charles II established a royal observatory at Greenwich in 1675, hoping to boost Britain's chances on the high seas by plotting the positions of various heavenly bodies as accurately as possible. Key amongst the equipment here was the transit telescope, angled to move only up and down, and from 1750 it was this telescope that defined Britain's prime meridian. A new transit telescope was built at Greenwich in 1850 by the Astronomer Royal George Airy. It was positioned a short distance away from the original, and this resulted in the Greenwich Meridian being moved nineteen feet eastwards.
Nobody other than seafarers and astronomers would have realised that this line existed, neither did they need to know. Local noon was just whenever the sun was overhead, which was easy because people never travelled very far away from home in those days. Then came the railways, and suddenly it mattered that noon in London was five minutes later than noon in Norwich but nine minutes earlier than noon in Bristol. Railway time soon became Greenwich time, and clocks in towns and villages across Britain were quick to follow suit.
In 1884 the International Meridian Conference was convened in Washington DC. Representatives of 25 countries met to establish a single meridian so that time and dates could be standardised across the globe. Almost everybody supported Greenwich, except for the French who had established their own meridien through Paris. But it was Airy's meridian in Greenwich that was confirmed as the world's official line of zero longitude and the basis of the new International Time Zone system.
In the 21st century global navigation now relies on the satellite technology of GPS. Accurate measurements from space have led to another slight shift of the globe's most important line, and the GPS meridian now lies 102.48 metres east of the old Greenwich meridian. This page has a go at explaining exactly why this difference exists, but it's mighty complicated. (And ah, so that's why my own little GPSdevice has been giving me what looked like dodgy readings in the Greenwich area).
After dark a green laser beam shines out from the Greenwich observatory along the meridian. It marks a very special line, and yet a completely arbitrary one. This is the benchmark of both time and space, the line from which all days begin and the only line on earth on which local noon is still noon. In the eyes of the world, Greenwich means time.