This is a critical value for estimating the rate at which a vehicle can accelerate. A light vehicle with a powerful engine will usually accelerate quicker than a heavy vehicle with a less powerful engine.
Here are a few examples, with the weight measured in kg and power in kW:
F-15 Eagle fighter plane: 5.99 kW/kg (Max take-off weight)
Concorde 1.66 kW/kg (Max take-off weight)
Formula 1 Racing car 1.000 kW/kg
Boeing 747-400 0.695 kW/kg (Max take-off weight)
Ducati 998R super bike 0.557 kW/kg (0.4 with rider)
Ferrari F50 0.315 kW/kg
Mercedes-Benz Brabus V12 0.241 kW/kg
Lamborghini Diablo 6.0 0.238 kW/kg
Jaguar XKR Coupe 0.137 kW/kg
VW Golf (standard model) 0.063 kW/kg
Mack CX 603 truck (cab only) 0.051 kW/kg (maximum engine size)
Olympic athlete 0.031 kW/kg (estimated)
The vehicles meant for high performance are the ones which display the highest power to weight ratios. The big truck is meant for sustained speeds over long distances, but not for repeated start/stop cycles
The idea comes from Newton's Second Law of Motion, which says, in essence, that acceleration is proportional to the motive force divided by the mass of an object.
Power, of course, is not simply force, but force multiplied by speed, so a scientist or engineer might want to look at the force applied to the vehicle (which involves gear ratios, and other factors) rather than the raw power output of the engine. Motoring enthusiasts, by contrast, tend to look primarily at the power output of the engine and divide it by the gross weight of the vehicle. This involves an implicit (and extremely optimistic) assumption that all the engine power can be transferred to the road surface during hard acceleration.
For most wheeled vehicles (cars, trucks and so on) the specifications list the power output of the engine. Whereas for jet aircraft, the specification is normally written in terms of thrust from the engines. In order to compare aircraft against cars, I have taken the maximum engine thrust and multiplied it by the maximum speed (or cruising speed, where applicable), in order to obtain a figure for power.
For the reasons above, official motor racing specifications tend to define limits on power output and also on the gearbox ratios and differential ratios.