The Dragon Balls are seven magic orbs about the size of baseballs, each with a number of tiny glowing stars in them, from one to seven. An ancient legend explains their powers: when the seven Dragon Balls are gathered together and the correct invocation is made, a mighty dragon appears and grants any one wish of someone who is present. Then, the dragon vanishes and the seven Dragon Balls scatter across the globe and turn into ordinary rocks for a year, before regaining their powers.
Eventually, in the manga and anime, we learn a little more about the Dragon Balls: they are artifacts created by Kami-sama (literally "god"), the mysterious supernatural being in charge of protecting Earth. When Goku meets Kami-sama he finds out why the Dragon Balls were made, although the explanation isn't so convincing (basically, Kami-sama made them to give humans something to strive for). Gradually we also find out that there are various rules that apply to the wishes. For instance, you can bring the dead back to life, but you can't bring the same person back twice. Further complicating matters, the second half of the series introduces a second set of Dragon Balls, the Namekian Dragon Balls, which have different rules and summon a different dragon. The anime-only storyline of Dragon Ball GT introduces still more Dragon Ball variants.
Bulma's quest for the Dragon Balls, for which she recruits Goku as her bodyguard, is what sets Dragon Ball in motion. However, the truth is that, in the grand scheme of things over the 42-volume graphic novel series, the Dragon Balls really aren't that important. Bad guys are always trying to get them so they can wish to rule the world or attain ultimate power, and the good guys are always trying to stop them. The only thing the good guys ever use the Dragon Balls for, more or less, is to wish their dead friends back to life. In short, they're a MacGuffin. Dragon Balls are the title of the series, but they're no substitute for good ol' martial arts training.