In the beginning of Toriyama's anime series Dragonball, Goku was just some kid with a tail living alone in the woods. His "grandfather", a master martial artist by the name of Gohan used to care for him, but had died when a giant gorilla-like beast attacked. The only thing he left behind was a Dragon Ball, which young Goku always guards fiercely. (In fact, much of motivation for various adventures in Dragonball are Goku trying to reclaim his grandfather's dragon ball.)

Goku is discovered by Bulma in the first episode, as she is searching for the dragon balls. As Goku's not about to give his up and makes a very useful bodyguard, they travel together for the beginning of the series.

Goku's key character trait is that he's completely naive -- he has no real common sense or tact. Early, this is cute, funny, and makes for good yet often crude jokes. (For example, it takes him a long time to be able to differentiate males and females by sight so he's continually grabbing at people's crotches and getting in trouble.) Later, this just turns him into the world's most powerful sucker.

In the course of his many adventures in Dragonball, Goku trains with Master Roshi, a.k.a. the Turtle Master, befriends Krillin, and eventually marries ChiChi. He also meets a whole host of other people, including Chiaozu, Tienshinhan, the Ox King, Yamcha, Oolong, Puar, Yajirobe, and in the end, Piccolo. (Piccolo is originally the series's "final villain", which explains a lot to some Americans who may have seen the first episode of Dragonball Z and been completely confused.) He also defeats an equally impressive number of bad guys, most of which either want the Dragon Balls or want to kill Goku because of his previous run-ins. In particular, the Red Ribbon Army and Lord Pilaf keep cropping up. In fact, the R.R.A. makes a return visit in Dragonball Z when a former leader develops some very nasty androids.

A few important things that we find out about Goku is that he actually killed his grandfather. Whenever he sees the full moon, he turns into a giant gorilla with amazing super powers (surprise, surprise). That is, unless someone cuts off his tail. Yet later we learn that this is because he's a Saiyan named Kakarot from an alien planet, sent to Earth to dominate it, basically. Obviously the Saiyan's plan didn't work, as Goku is terminally nice.

And the story just keeps going on from there... By the beginning of Dragonball Z, Goku has a son, Gohan (or Son Gohan), who starts taking a lot of the focus. In Dragonball GT, Lord Pilaf (again) finds the seven Dragonballs in one place (while the Z Team is a stone's throw away and not paying attention) and summons Earth's dragon (Shenron). Goku interrupts them and Pilaf mistakenly wishes him back to childhood.

For anyone remotely interested in Goku's history or the series, I highly reccomend you watch the original Japanese Dragonball series, as it's much better than Dragonball Z, which in turn is better than the Americanized dreck on Cartoon Network.