ShenLong - This being the Earth Dragon God, was created by Piccolo's good half (Kami Sama - God). Piccolo came from the Namek family, who conceived the original Dragon (Porunga). So, when he came to Earth as a child (due to terrible weather problems in Namek, Piccolo was sent by his parents), he had the urge to create a similar dragon (he did not do this consciously, for he had no memory of his planet).
In the beginning of Dragonball we are told that once, a long time ago, there were no Dragonballs, you just needed to ask your wish to the Dragon, who would sit at a palace. But due to Man's greed and violence, God decided to conceive a way that only the most worthy could have their wishes granted. So he created the Dragonballs, which would be dispersed all around the Globe, making it difficult to gather all of them (especially without a Dragonball Radar, which was first constructed by Bulma). Shenlong, appears, when all the Dragonballs are united and the words "Ide yo, Doragon!" (Come out Dragon) are said, after a big "light show" the dragon comes out and grants the first wish asked of him (no matter, who says it).
Though, being said in the beginning of the series that all your wishes could be fulfilled, the Dragon had one limitation (besides only giving one wish!) he can't grant the same wish twice. This is a bit of a problem (especially in the second series), as deaths are common among the heroes, and nobody could be revived. Despite of some stranger requests (like the first time the dragonballs were gathered, where Oolong asked for some panties), in general, the more frequent requests are Immortality (normally the request of villains) and bring people back to life. The power of the dragon relies on the power of it's creator (this seems a bit of a problem, because if so, why can't Piccolo bring people back to life? - the Dragon seems to be a parallel manifestation of his own power; example: a man can create a Formula 1 car, but he, himself cannot run at 300Km/h ) and if the creator dies the dragon dies too (making the death of Piccolo a problem).
During the Cell Saga, Piccolo and Kami Sama finally fuse (this being a Namek only fuse) in order to fight Cell. So a new God is needed, and Dende is the chosen one; after some time in the job he decides to give the Dragon a power up, making him able to grant 3 wishes.