When jump-starting a vehicle, one first connects the positive and negative terminals between the two cars: positive to positive and negative (good battery) to a bolt head on the vehicle with the dead battery. Next comes the actual jump-starting part - turning the engine of the vehicle with the dead battery. My question is does the the engine of the car with the good battery have to be running before one attempts starting the engine of the vehicle with the dead battery, and if yes, then why?
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Your description of connecting the two cars is correct. I want to emphasize making the last connection to the "bolt head" you mention. Any large piece of unpainted metal will do. The reason for this is to keep the resulting spark away from the battery. If they battery has been venting fumes, the spark can (although very unlikely) cause them to ignite and explode.
Back to your question: It is better to have the "good" car's engine running, because its alternator will be able to provide current to the "dead" car in addition to keeping the good car's battery charged.
Also, when the dead car actually cranks up, it's own alternator can cause a significant spike into both car's systems. Now that cars rely so heavily on digital electronics, this spike can damage the controlling computer. It is wise to turn on the parking lights (not the headlights, because they are too power-hungry) of the both cars before starting the bad car. The surge will be dampened by the light bulbs, which will flow more current when the voltage increases.
Both of these events (the exploding battery and the fried computer) are really unlikely, and people jump cars without these precautions all the time. But, in this case, it doesn't hurt to be safe :)