DC current is direct current, which means electricity only flows in one direction; AC current is alternating current, which means it flows in both directions. Batteries run on DC current, while AC current is what comes in through an electrical plug in the wall.
One reason alternating current is used for electricity in the home is because it is easy to change the voltage of the power with a transformer. In the United States, what comes out of the wall is 120 volts. Alternating current switches back and forth at a rapid rate; in the U.S. it switches 60 times per second.