Well, we'll start here since these objects are the title of the manga. The Dragon Balls are legendary crystal spheres, orange in color, of which if one collects all seven of them and invokes the name of the "Shenlon" (Dragon God), the finder is granted one wish.

After the wish is granted, the balls fly up into the air and disperse across the globe, turning into stone until they reactivate themselves in a year.

The seven balls, each containing one to seven stars in them, were created by Kami-sama to instill hope in the world. However, they were often misused in the past.

We later learn that there is another set of Dragon Balls, on Planet Namek. It turns out that Dragon Balls originated there and that the idea was taken to earth by Kami-sama (who turns out to be a Namek alien). The difference is that the Namek Dragon Balls are much larger than earthly ones, the wish-giver is called "Porunga," and the finder receives 3 wishes. However, while with the earth Dragon Balls one can revive as many people as one wants with one wish (provided that they haven't been dead twice), the Namek balls restrict the number of people to one per wish (although the number of times one has been dead doesn't matter).

Although the Dragon Balls seem all-powerful, there is an important qualification to the wishes, the Shenlon/Porunga can't grant wishes that are beyond their powers. For example, they can't stop enemies from invading the earth if the enemies are stronger than Shenlon/Porunga. However, the GT "Ultimate Dragon Balls" seem to be a different matter all together.

Here is some information on all the different types of DragonBalls, the creators, and how they work:

Kame-sama's Dragon Balls

The original set on Earth that Gokou and his friends have been chasing their whole lives. They are like big orange marbles. And in the middle they have red stars in the middle. There are seven dragon balls, each one has a different amount of stars in the middle (1-7 stars). Kame-sama made these not too long after spilting with Piccolo Daimao. After a wish is granted, the Dragon Balls spread across the planet and turn to stone for one Earth year (365 days), the Dragon Balls cannot be used to summon Shenron until they have reverted from stone.

The following is required to grant a wish from Shenron:
1. Must gather all 7 Dragon Balls of the same set.
2. Chant must be spoken - anything along the lines of "Shenron come out."
3. The wish cannot exceed the power of the creator of the Dragon Balls. - If Kame-same cannot do it, then either can Shenron.
4. A person can only be wished back to life once using the Dragon Balls (I don't know if someone is wished back using another set of Dragon Balls, I'm assuming you can't).
5. The wish must contain "I wish...", "Grant me the power to...", "Make me...", or anything along those lines.
6. Multiple people can be brought back to life using the same wish.
7. Only one wish per summoning of Shenron.
8. You cannot wish for more wishes.

Dende's Dragon Balls

Since Piccolo and Kame-sama merged/fused into one being again (Cell Saga), Kame-sama no longer existed, and either did his set. Gokou teleported with Dende from New Nameksei to Earth then Mr. Popo and Piccolo taught Dende how to make his own set. Dende's balls, err... I mean, the set that he created are the same as Kame-sama's except an extra wish. Someone once said that Dende's set has white stars, but this is false. Shenron looks exactly the same as Kame-sama's. After the wish(es) are granted, the Dragon Balls spread across the planet and turn to stone for one Earth year. If only one wish was granted, they remain as stone for only 4 months.

The following is required to grant a wish from Shenron:
1. Must gather all 7 Dragon Balls of the same set.
2. Chant must be spoken - anything along the lines of "Shenron come out."
3. The wish cannot exceed the power of the creator of the Dragon Balls. - If Dende cannot do it, then either can Shenron.
4. A person can only be wished back to life once using the Dragon Balls (I don't know if someone is wished back using another set of Dragon Balls, I'm assuming you can't).
5. The wish must contain "I wish...", "Grant me the power to...", "Make me...", or anything along those lines.
6. Multiple people can be brought back to life using the same wish.
7. Only two wishes per summoning of Shenron.
8. You cannot wish for more wishes.

The Namekseijin Dragon Balls

The first set made long ago on the oringinal Nameksei by Suruchsai . The Namekseijin Dragon Balls are much bigger, bigger than a basketball, therefore Porunga is more powerful than Shenron, and more wishes!! The dragon, named Porunga, doesn't look anything like Shenron, he isn't as long, but is very muscular. The Dragon Balls are still orange with red stars. After the wish(es) are granted, the Dragon Balls spread across the planet and turn to stone for one Namekseijin year (about 183 Earth days). I am currently unsure how long the Dragon Balls remain as stone if not all the wishes are granted.

The following is required to grant a wish from Porunga:
1. Must gather all 7 Dragon Balls of the same set.
2. Chant must be spoken in Namekseijin language - anything along the lines of "Porunga come out."
3. The wish cannot exceed the power of the creator of the Dragon Balls. - If the creator couldn't couldn't do it, then either can Porunga.
4. A person can be wished back to life as many times as needed using the Dragon Balls.
5. The wish must contain "I wish...", "Grant me the power to...", "Make me...", or anything along those lines in the Namekseijin language.
6. Only one person can be brought back to life per wish.
7. Only three wishes per summoning of Porunga.
8. You cannot wish for more wishes. (Isn't three enough?! You greedy Otaku.)

Kamiccolo's Dragon Balls

(Kamiccolo is Kame-sama and Piccolo Daimao before they split, that's not his real name, the fans gave him the name for lack of anything else) The set made by Kamiccolo, before he spilt. These only appear in DBGT. Kamiccolo's set have black stars in the middle instead of red, to reflect the evil in his heart. This set is also commonly known as the "Black Star Dragon Balls." Shenron is HUGE. His head is like half the size of Kame-sama's floating palace and he's red, not green like the other Dragons. Since Kamiccolo was mainly evil, the evil was reflected in his Dragon Balls. After a wish is granted, the Dragon Balls spread across the four galaxies, if they are not all gathered together again within one Earth year, the planet which the last wish was granted will be destroyed.

The following is required to grant a wish from Shenron:
1. Must gather all 7 Dragon Balls of the same set.
2. Chant must be spoken - anything along the lines of "Shenron come out."
3. The wish cannot exceed the power of the Dragon, but this Shenron is mad powerful and I don't believe that there is a wish that he cannot grant.
4. A person can only be wished back to life once using the Dragon Balls (I don't know if someone is wished back using another set of Dragon Balls, I'm assuming you can't).
5. The wish must contain "I wish...", "Grant me the power to...", "Make me...", or anything along those lines.
6. Multiple people can be brought back to life using the same wish.
7. Only one wish per summoning of Shenron.
8. You cannot wish for more wishes.