A SIMPLE EXPLANATION
Electrical current is the flow of electrons through wiring and electrical components. Flowing electrons (current) can perform work such as making a light bulb glow, or turning a motor.
We've found that we can get flowing electrons to do this kind of work when they flow consistently in one direction.
We've also found ways to get flowing electrons to do the same work when they flow momentarily on one direction, then turn on themselves and flow in the other direction, then switch back to flowing in the first direction, and then reverse themselves again repeatedly.
Electrons that flow always in one direction create what we've designated as Direct Current (DC). The batteries in your flashlight, cell phone, iPod, and automobile all supply direct current.
Electron flow that keeps switching directions creates what we've designated as Alternating Current (AC). The electricity coming from the power lines into your home is alternating current.
[SIDE BAR: Because AC current switches directions 60 times per second, it is referred to as 60 Hertz. In European countries, it is often 50 Hertz-the reason why you sometimes have problems with personal appliances you take overseas.]
We have both AC and DC because we can make them both work, and each one is easier to produce in different circumstances.
To generate electricity in a power plant, we've found the most efficient way is to use some other type of power to turn a generator. The other types of power are flowing water, or steam created by burning coal, natural gas, or from nuclear fission.
The internal design of a generator naturally produces alternating current. Since it's easier to produce alternating current than direct current this way, and we can use it, that's what we do. And that's what gets sent to our homes.
When we want to store electricity for use later, we use batteries. The physical properties of batteries only produce direct current. But since we can use it, that's what we do.
For the most part, light bulbs don't care if the electrons flow always in one direction or if the electrons are constantly switching directions. The electron movement still makes the filament hot and it glows to produce light.
Electric motors are a little more finicky. AC motors need AC current and DC motors need DC current. That's usually not a problem because you're not often going to try to run your blender off of your car battery, or four D cells.
When you do have devices that can work on batteries as well as when plugged into a regular house outlet (a radio or a laptop computer, for example), the circuits inside the device are usually designed to work on DC, and will convert the AC electricity into DC when it's plugged into an AC outlet.
The same thing happens when you plug in your cell phone battery charger (or any other battery charger for that matter.) That little black box on your battery charger contains rectifiers and transformers that convert the high-voltage AC current from your outlet into low-voltage direct current equivalent to battery current. This not only is sufficient to operate your device, it recharges the battery as well.
So to summarize, AC current is produced in a power plant and delivered to the electrical outlets in your home. DC current is produced by a battery and powers your portable devices.
(Note, as always, there are exceptions to the rules. For example, when your car engine is running, it runs a generator to produce AC current, like a power plant. You car can't use the AC current so it is converted into DC current to run your vehicle's systems and recharge the battery.)
(And yes, I know "AC current" is as redundant as "IRA account. But as Walter Cronkite would have said, "That's the way it is!")