All of the answers to the question are correct. Basically, when Edison was first developing electrical generators on a power-grid scale, he hired Nikola Tesla as a protege, and Tesla, it is claimed, used the principles of alternating current and polyphase power to greatly increase the efficiency of electrical generators, which by Edison's original designs produced DC.
Basically, the big deal is that AC requires less work for more power (i.e. it is more efficient to generate). Think of an electric current in terms of a closed, pressurized water loop; water is put under pressure by some power source, which causes it to flow through hoses to some gadget that can use the flow of water to do mechanical work. The water, its energy spent, is then drawn back to the power source.
DC would be the equivalent of putting pressure on the water in one direction only, either by feeding it from a tank (similar to how a battery would work) or by using an impeller or other rotating pump (similar to a generator). Such a pump would move water inefficiently, as the pumping mechanism cannot be watertight. A one-way reciprocating pump would be watertight but would not move water constantly, which may be overcome (as in AC to DC converters) using a reservoir that will store additional pressure and then feed it in to the system while the pump is on its "backstroke". Any way you slice it, except in the case of a tank (battery), there is wasted effort in producing the current.
AC, by contrast, would be the equivalent of using a simple reciprocating pump to force water one way, then the other. As long as the devices expect the flow of water to reverse itself (or don't care), the design of the generator can be much simpler, and more efficient. The reasons for the efficiency gains are a little different when you do away with the analogy, but the analogy itself holds pretty well.
AC also has a few tricks up its sleeve that DC simply cannot replicate, which make it preferable to DC for large-scale applications. Perhaps most important is the ability to be "stepped up" as well as "stepped down" using a transformer. DC can only be "stepped down" using resistors, which basically transform electrical energy to heat and thus cause you to waste a lot of energy. Polyphase power, seen in the U.S. as 3-phase power, is more a solution to a problem of AC than a benefit (Three-phase AC allows the power grid to have near-constant overall voltage, overcoming the non-constant voltage of a single AC waveform, while using less wire than would be required to efficiently transfer the same overall power in a single waveform), but it provides the beneficial side effect of being able to "add" phases to each other for the same available current. In split phase, voltage is doubled while in 3-phase, voltage is multipled by √3. This is why residential is 120/240V (120 * 2) and commercial 120/208 (120 * √3).