Usinf your DC analogy, current flows "out" the black and "in" the red, or vice versa. If you picture your supply as a large transformer, there is 240 volts between the two "ends" of the transformer. If a load, such as a heater, is connected to these two points, current flows.
Now, the neutral wire is connected at the center of the transformer, so 120 volts would be measured between the white lead and either red or black. On a time diagram, the two voltages are 180º out of phase, but this is not important in house wiring. Every load is considered a single phase load.
In house wiring, the neutral wire is connected to ground at the panel. In a 120 volt circuit, current flows in the neutral wire, so it cannot be part of the safety ground. In a device like a wall heater, all that is needed is the red and black. A ground (bare) wire should also be provided to ground any metal junction boxes or cabinets.
In a stove or dryer, the main loads are 240 ( heaters ) but there may also be some 120 volt components such as timer clocks, drive motors. So the white wire is needed.
In older codes, the white wire was allowed to serve as the safety ground, on the theory that the load was so balanced that little or no current actually flows in the white. Today, most codes do require a 4-wire plug to stove or dryer.