Much like DA01 said, a ceiling fan generally acts to circulate air within a particular space of the home. This evens out the temperature of the air in the room by preventing "stratification" (where the air settles into noticeable "layers" so it's warmer at head height than at the floor), and also provides an illusion that the air is cooler by constantly blowing away the "bubble" of insulating warm air that your warm-blooded body will maintain around itself (also speeding evaporation of sweat).
Unlike your forced-air ventilation, the ability of a ceiling fan, or any non-ducted fan, to "push" a column of air into a different area is limited by the buffering effect that the standing air will have on the air pushed by the fan. A forced-air system uses ducts, within which air ideally only moves in one direction; supply vents carry conditioned (heated/cooled) air out to the home, while one or more returns draw in the air that will be heated/cooled. This generally forces the air to move in a large loop through your home from supply ducts to returns through the rooms.
A ceiling fan or other "open-air" ventilation doesn't have these ducts; the air that is pushed forward by the fan must be replaced by air moving within the same space as the pushed air. This conflicting movement of air creates eddies that reduce the effect of the fan as you move further from it. Within an enclosed space like a room, this effect isn't so bad, but trying to move air through a hallway, or up or down a staircase, will not be effective.
Your dual-zone HVAC is your best tool for achieving the proper temperature on each floor of your home. You'll just have to remember that warm air rises, cool air descends, so in warm weather your upstairs A/C will have to work harder, because it will be losing some of the cool air it produces to the downstairs area and will receive the warm air from both floors. The opposite is true in winter; the downstairs heater will effectively be heating both floors, and will have to work harder to heat the downstairs as all the cool air will work its way down.
Instead of trying to mix the air between floors, you might actually have better luck trying to prevent the two zones from exchanging air, using a door, curtain or other insulating break at the top of the upstairs staircase. By doing this, you reduce the amount of cool air that the upstairs will lose and the amount of warm air it will gain. This will allow the upstairs zone to cool its own space without also having to cool the downstairs, and vice-versa for heating in winter. Of course, this will only work if you have separate air returns for upstairs and down; if you only have one air return, then air will be actively pulled toward that return, wherever it is, and regardless of what's in the way.
Best Answer
The general idea is
Air blowing down = cool
Largely this is a felt effect, because the air is moving over you, i.e. wind chill
Air blowing up = warm
Hot air rises. Pulling colder air up and out pushes it off the ceiling and towards the walls and down. Because you don't feel the effects of the air movement directly it doesn't add to the chill effect. This one has less noticeable effect because it's trying to mix the air. It's also greatly influenced by how much insulation your ceiling has (less insulation = less heat for the fan to push). Still, it's better than nothing if you're looking to milk efficiency. It's also less effective in larger rooms, where the walls are not close enough to help force the hot air down.