Power is a technical term. It does more than sells cars and, in fact, can aid in winning races. Unfortunately its true nature is sometimes poorly understood, even by car enthusiasts. Power is also often put in an unnecessary rivalry with another engine output metric - maximum torque.
There are a lot of explanations on the Internet and elsewhere that try to explain what's what. Some of them are right, some not. Many aren't very clear. What is clear, is the physics behind cars, and I hope that, by providing an explanation based in physics, I can clear up misunderstandings that anyone might have on what horsepower is and what it really says about a car or engine.
In physics, power is the rate of change of energy. A car in motion has a particular kind of energy known as kinetic energy (KE). KE is related to motion, the faster something goes, the more KE it has. The exact relation is KE = .5 * mass * velocity2.
Velocity is something that most people understand, and it's something that a driver wants to maximize when racing. Using the equation above, it's possible to determine the amount of KE carried by a car at a certain speed if we know its mass.
A 1000 kg car traveling at 50 m/s possesses 1,250,000 joules of energy. Note that units are important. Mass is in kilograms (kg), you can't use Newtons or pound-force (as they are weight) unless you use their mass equivalents - the Newton-mass and pound-mass, as long as you include the proper correction factor.
The KE is important and can provide a lot of information. When a car tries to stop, the brakes need to remove all the KE from the car by converting it into heat. The more KE, the more the brakes have to work. It's pretty clear that having a lighter car is better for your brakes. But none of this has to do with horsepower...
Let's imagine the same 1000 kg car from before traveling at 50 m/s as it starts to accelerate. It would be useful to calculate how fast it can accelerate. This can be determined from torque if the gear ratios and tire sizes are known, but goog-ling this information might take minutes!
Instead of using torque, it's possible to use power, and this makes things much easier. Now the gearing and the wheels don't matter at all. Power provides a method to determine the acceleration of a car, whether it has mile wide gear spacing and bicycle tires, or a gear limited speed of 5 mph and monster truck tires.
For simplicity's sake, assume that the engine makes a constant 500,000 Watts (units are important) of power at every RPM. The engine will simply add 500,000 joules of energy to the car every second since a Watt is equivalent to a joule per second.
So the car starts at 1,250,000 Joules or 50 m/s. One second later it will have 1,750,000 joules and its speed will be 59 m/s. At 2 seconds from starting, KE is 2,250,000 and speed is 67 m/s. It's clear that even though power is constant, the acceleration is not. 500,000 W averaged 9 m/s2 acceleration from time zero to one, but only 8 m/s2 from time one to two. This is because 500,000 joules becomes a smaller and smaller percentage of the total KE. The faster the car goes, the harder it is to make it accelerate.
Thinking in terms of energy might seem a little abstract however. Force is a more familiar term, and fortunately, power and force are related. To see this relationship, requires a deeper understanding on energy.
Energy, in general, is the ability to do Work, and Work is force exerted over a distance. 1 Newton of force exerted for 1 meter of distance is 1 joule of work, or 1 joule of energy. Power is the rate of change of energy. For those familiar with calculus, {Power is the derivative of energy. Don't worry, I won't mention calculus again...
Displacement (Distance), Velocity, and Acceleration are related in the same way that energy and power are. Acceleration is the rate of change of velocity, which is the rate of change of distance. Going back to the Work/Force relationship (Energy = Work = Force * Distance), let's turn energy into power by using rate of change (RoC).
Power = RoC of Energy
Power = RoC of Force * Distance
Velocity = RoC of Distance
RoC of Force * Distance = Force * RoC of Distance
RoC of Force * Distance = Force * Velocity
Power = Force * Velocity
Now if the engine's power is known, so is the force pushing the car. Force and Acceleration are already known.
F = ma
Acceleration = Force / mass
Acceleration = Power /(Velocity * mass)
Unlike the KE method used before, this new equation can calculate the acceleration at any point in time. Going back to the example car at 1000 kg with the 500,000 W engine, acceleration is 10 m/s2 at time zero, 8.5 m/s2 at time one, and 7.5 m/s2 at time two.
Horsepower tells you everything you need to know about how the car will accelerate. Power and torque are really the same thing as far as the car in concerned, but horsepower is far easier to use, and this is why it is such a popular metric.
The power and force relationship also allows for other helpful calculations, like those involving drag. Drag is a force, but using the equation relating power to force, it's possible to convert drag force into drag power.
Drag power can be thought of as the opposite of an engine. Instead of creating power, it absorbs it and slows the car down. To see the effect of drag on a car's acceleration, add the drag power to the engine power. Remember that drag power will be negative in this case. Since drag gets bigger with speed, at some point the engine power + drag power will equal zero. At that point, the car would be at its maximum speed.
All of this still leaves one question, where did the horsepower vs torque argument come from? It's actually a result of poor terminology. The heart of the argument lies with the engine's power-band, and not the peak values of horsepower and torque or the individual properties of those two quantities.
Peak refers to the maximum values of power and torque. These numbers can be misleading because they only occur in a narrow RPM range. As long as an engine does not have continuously variable transmission (CVT) with no distinct gear ratios, the power and torque over a range of RPM is more important than the peak values.
For example, a 500 hp car can easily be quicker than a 1000 hp car. If the 500 hp car produces at least 90% of peak power within 2000 RPM of the peak power RPM, it will easily outrun the 1000 hp car that only produces 125 hp of power everywhere that is off peak RPM, and doesn't have a CVT.
Going back to the horsepower vs torque debate, a torque-rich engine is one that produces torque over a wide range of RPM, or one that produces large amount of torque (and thus horsepower) at low RPM. This gives the car good acceleration from a low speed even if the engine is not spinning very fast. This is great for road cars where the RPM is usually low and there is a lot of stop and go driving.
Conversely, a horsepower-rich engine is one that forgoes torque at low RPM in order to produce more torque (and thus horsepower) at high RPM. This is actually preferred for race cars, and the reason should be obvious once the relationship between power and torque is understood. Power is proportional to torque multiplied by RPM. In other words, 500 units of torque produces more power at 6000 RPM than it does at 5000 RPM. The "horsepower" engine produces more power overall, and this makes it faster.
This isn't to say that low end torque isn't important for race cars though. Depending on the track, a wide, torque-rich power-band can prove advantageous. A torque-ish engine requires less downshifting then a peaky engine, and it also allows the gears to be spaced further apart or for unneeded gears to be dropped completely to save weight.
In the end, power isn't all that complicated and it's not some superfluous number. It's a very helpful performance metric and should be seriously considered in any comparison between sports cars.