HorsePower is a measurement of how many pounds of pressure the engine produces. I think you get it by multiplying the amount of pressure each cylinder produces by the number of cylinders. I think it tells you very little.
Torque is a measurement of circular pressure, that is, how much pressure is exerted in a turning fashion. This is calculated in pound-feet, and is calculated by multiplying the pounds of pressure by the distance from the axis.
To make that clearer: Imagine you have a rod that you wish to turn. You are going to turn it by exerting pressure on another rod which is attached to it perpendicularly. (shaped like a T, you are moving the top of the T to spin the upright). The amount of pressure you exert on the top of the T will have more effect if you hold it farther away from the shaft, because of a lever effect. This is why you are able to remove your lug nuts with a wrench but not with a screw driver.
Therefore, to calculate how much pressure is being exerted in a circular way on the shaft, you need to multiply the pressure by the distance from the shaft–“pound-feet”.
In a car, the pressure is exerted by the pistons on the drive shaft. You need to multiply the pressure exerted by the distance from the center of the drive shaft to arrive at torque.
Thus, torque is an important number for measuring how much weight your car will be able to move, though not really for how fast it will do so.
I have no idea anything about pickup. Pickup can usually be measured by numbers the car companies report as 0-60.