You can change the voltage of AC power with a transformer. You can't do that with DC.
It is more efficient to transmit power for long distances at high voltages. The same amount of power can be sent over thinner wires, with lower resistive losses, than would be possible at low voltages.
Power transmission on wooden poles might be 14,000 - 28,000 Volts, AC. On big steel towers, 350,000 to possibly 1,000,000 Volts AC. (There are a few transmission lines that operate at high voltages, DC. I'm going to ignore them in this answer.)
Those high voltages are too dangerous for home use, of course. Power companies use transformers to step it down to 120 and 240 Volts for home use.
Your home electronic devices need DC to operate. That's just the nature of transistors and integrated circuits.
EDIT, 2-AUG-13: Seth: Thanks for the addition. I'll just comment that the reason for high-voltage DC is that, for very long distance AC transmission, the power lines start working like antennas. Work it out: for 60 Hz power, a power line starts looking like a quarter-wave antenna at around 750 miles.
I hate it when that happens. :-)
With alternating current (AC) you can use transformers to raise and lower the voltage. With direct current (DC) you can't unless you use an inverter to convert it to AC, change the voltage with a transformer, then convert it back to DC with a rectifier at a large loss of amperage (electricity's ability to do work). Back when AC first popped up to challenge DC there were no electronic means to convert DC to AC or vice versa, but only electro-mechanical, so the amperage losses would have been even greater.
With no way to adjust the voltage with DC, it could only usefully be transmitted a couple miles from where it was being generated because of resistance in the wires lowering voltage with distance from the power plant, which meant that in a large city you would need power plants every 4 miles or so. With AC you can have a power plant out on the fringes of the city, or generated at a dam tens of miles away and still be able to transmit that power.
With AC you have transformers at the power plant that are raising the voltage for transmission over high tension power lines. When the lines get to the neighborhood where the power is going to be used there's a power sub-station (with large transformers) that is lowering the voltage for use within the neighborhood. Usually the voltage coming from the sub-station is higher than what you would use in your home, so closer to the sub-station the trashcan-style transformers on the poles are being used to adjust the voltage downward to a home useful voltage; while farther from the sub-station where the voltage has dropped lower than what is home useful because of line resistance, those trashcan-style transformers are raising the voltage to a home useful voltage.
Joe's answer covers most of it, but DC is used now for some high voltage transmission. The main reason goes back to Edison vs. Westinghouse. Edison's system used DC and Westinghouse used AC. Westinghouse's system won out because it is MUCH easier to transform voltages with AC. Back then, the only way to transform DC voltages was to convert to AC and use a transformer (which only works on AC), then convert back to DC. This is costly and inefficient. As Joe points out, voltages must be stepped up in order to transmit efficiently over long distances, but must be stepped down to be safe for consumption.
Nowadays, it is possible to directly step DC voltages up and down without a transformer. But all the legacy equipment and consumer devices would make it completely impractical to change over an existing AC system to DC. Furthermore, the equipment for changing DC voltages is still more complex and thus more costly than for AC. One of the big problems with high-voltage DC is that switching a circuit on or off creates a much longer arc than AC, since with AC the voltage drops to zero for a moment during each cycle, but not with DC.
Transmission of AC is easy
Semiconductor uses DC low operating voltage to work
AC can be easy converted into DC but no vice-versa
What are the advantages of using A/C?
