Lets forget about rpm, torque peak, and engine.

From physic point of view, in layman term:

Torque = ability to carry load
High torque means high load bearing ability. Low load bearing ability means a vehicle will immediately feel sluggish when a load is added.

Horsepower = ability to accelerate[/u]
High horsepower means better acceleration. For a vehicle with same load, high horsepower means it accelerate better.

Ok, now lets get complicated. A vehicle with small torque but huge horsepower has amazing acceleration. But any additional load will make significant impact on its acceleration, because the inability to handle load (due to low torque). For example, motorbike.

Then, a vehicle with huge torque but small horsepower has low acceleration, but excellent load handling. By load handling means, the vehicle won't feel much different when a load is added. For example, bulldozer.

When it comes to good engine management (and driving experience), race car driver generally drive the car at high rpm near peak hp. While truck driver generally drive the vehicle around peak torque.

Now lets take a real scenario. Say a car with peak torque at 4k rpm and peak hp at 6k rpm.

During uphill drive, when engine speed drop below peak torque (say 3k rpm), the engine immediately feel sluggish. Hence we downshift, accelerate (rev) the engine to higher rpm, say to 5k rpm (higher than peak torque). After that, when we shift the gear back, say 4k, we immediately feel the engine power returns.

The sames goes to why tires are more likely to burn (skid) when we accelerate from standstill at peak hp.

Hmm, I have to disagree with you. (No offense, but this is what phd section is all about: argument and discussion ) In fact, motor is a perfectly good example of how engine can be so different. I'll drop the efficiency factor, since this is a discussion about torque and horsepower.

Torque
Torque is the resultant of inertia x angular acceleration.

Horsepower
Horsepower (or power) is the rate of energy release. High horsepower means high rate of energy release. Like explosion.
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Motor
Motor has a constant (or near constant) torque with respect to RPM. Torque depends on voltage. While its power (or horsepower) depends on current draw (assume constant voltage). A motor with low resistance has high power rating. Motor power rating is also constant (due to constant resistant).

For example, take table fan. Without the blade, when we turned it on, the motor spin to its maximum speed almost immediately. And when we put on the blade, (higher inertia), the motor take some times to reach its maximum speed. Probably you also notice that fan with blade does not spin as fast as without. This is because air resistance increase as the speed increase. My last post use the assumption of fixed torque and power over the range, i.e. like a motor. Or more precisely, the instantaneous response (if you take dynamic, you know what I mean).

Engine
Engine, on the other hand has variable torque and horsepower. This is because its power input varies. (Motor has fixed power supply). Everyone knows engine runs due to small explosion in combustion chamber. And this combustion is not limited. (because someone "pedal to metal ").

In layman term, with fix fuel supply (maintain pedal position), a vehicle can travel at 100km/h (2500 rpm) on straight road, or 120km/h (3000 rpm) going downhill, or 70 km/h (2000 rpm) going uphill. Despite the different in speed and rpm, the engine actually consume same amount of fuel (in reality this might varies a bit, but this assumption is still generally true). What happen is that, the engine require less fuel to produce same amount of torque (during downhill) or more fuel during uphill.

Horsepower, on other hand, is the ability of the engine to response to fuel supply. (rate of energy release). An engine with high horsepower has the ability to response immediately to fuel, thus able to change its torque. By the chart, it simply means the ability to rev. If you notice, a car rpm increase faster when it approach peak hp, and fastest at peak hp. (We are unlikely to admit that it is fastest at peak hp. This is because hp drop after the peak hence it appear slow). Engines with VTEC, VVTI, CPS has higher peak hp at high rpm, this means driver will observe that the acceleration is highest when it reach peak hp.

What happen to campro engine, it has relatively acceptable hp chart (higher rate of energy release), which means angular acceleration (the speed rpm needle increase) is fairly constant. Because horsepower means the ability to change its torque, looking from the graph (campro), at 2000-3500 rpm, it means the ability to decrease its torque. This also mean it is less efficient in this interval. No wonder why campro owner complain about sluggish engine response. This effect is magnified during uphill. The dip in torque means it require even more fuel than other vehicle to carry itself uphill.

Back to motor, its high torque, low horsepower means it is good for road cruising, at constant speed. While for acceleration, the high horsepower engine comes in. The next generation of hybrid car will see the evolution of small cc engine (with high horsepower) and high torque motor.

How it is implemented, that require a new chapter of applied physic.

Let's just say my work require me to be familiar with it. (Does that means I can publish book already? ) Motor has high starting torque, (because it has constant torque curve). While engine has zero starting torque. Which is why we have a motor starter for all engine. Starter motor provide the torque to initiate the rotation in engine, but if engage with gear, it can't maintain the engine rotation because of huge load. Joey mention about motor, hence I try to explain how it is different than engine.

I think it's hp that power a vehicle out of corner, and torque is what keeps us going on straight. Brake horsepower, is different that horsepower. Indeed, horsepower and torque is inter-related. By definition, horsepower is derived from torque. But with engine, the ability to manipulate its input (fuel, in this case), and the torque/power curve makes it unique and difficult to explain.