In its original form, Dragon Ball is a manga (Japanese comic) written and drawn by Akira Toriyama from 1984 to 1995. Published by Shueisha, one of the three biggest Japanese publishers, it was the most popular series in Weekly Shônen Jump, a 400+ page comic anthology which features about 20 ongoing serials by different artists.
Dragon Ball is, basically, a martial arts story with elements of fantasy, science fiction and comedy. The hero, Goku, is a boy with a monkey's tail (or not, in the live-action version) who is raised in the woods by a martial artist. Bulma, a girl on a quest for the seven magic Dragon Balls (a treasure which can grant any wish), accidentally discovers the guileless Goku and introduces him to civilization. Over many adventures, Goku travels around the world, develops his already prodigious fighting skills, and saves the world from evil martial artists many times over.
This is the basic formula: lots of martial arts, lots of training sequences, a few jokes. (Sometimes dirty jokes.) Whether Goku's opponent is the green-skinned Great Demon King Piccolo (his first major opponent, played by James Marsters in Dragonball: Evolution), or the alien mercenary Vegeta (presumably next in line for the sequel), or the artificial life form Cell, or the genie-like magical pink blob of doom Boo, the structure is the same. New bad guys show up, and Goku must defeat them (as much out of a love of a good fight as a desire to save the planet); if he's not strong enough, or he loses the first round, he hits the gym and soon he's buff enough to have a fighting chance. Rinse and repeat for 14 pages a week, once a week for ten years, and you have a 42-volume, 8,000-page graphic novel series.
This Dragon Ball formula became the model for a successful shonen (boys') manga, inspiring such little-known works as Yu-Gi-Oh! and Naruto. But it wasn't always intended as an epic, and if it seems to have lots of quirky elements (such as Bulma's blue hair, or the magic nyoi-bo staff which Goku sometimes fights with, or the titular Dragon Balls themselves, which really aren't even that important in the manga), it's because the series changed a lot over its 10-year run. When he started Dragon Ball, Akira Toriyama was best known for his previous hit, the 1980-1984 mad scientist comedy manga Dr. Slump. Dragon Ball was also conceived as a comedy, or comedy-adventure, albeit based on Hong Kong martial arts movies rather than the science fiction genre. But readers reacted more to the action elements than the comedy, and so, with the guidance (or pressure) of his editors, Toriyama gradually de-emphasized the humor elements (such as the talking animals, which aren't in the movie) and emphasized the fighting and melodrama. The resulting hit combo was spun off into anime, video games, and merchandise, and made Shônen Jump magazine the manga equivalent of DC and Marvel put together; at its peak in the early '90s, before the magazine market started its slow worldwide decline, it sold 6.53 million copies per week. As for Dragon Ball, it was rated the #3 manga series of all time by Japan's Agency for Cultural Affairs .
The anime was first translated for English syndication in 1995, although it didn't become a hit until it started appearing on Cartoon Network's Toonami programming block in 1998, where less intense censorship allowed the characters to really beat each other up like in the original Japanese version. The manga was translated by VIZ and printed as two separate series, Dragon Ball and Dragon Ball Z... which leads into the next question.