Goku is a Saiyajin whose native world is the planet Vegita. He was sent to Earth in a space capsule right after he was born so he could conquer the planet. However, he hit his head and lost his "programming," thus becoming the kinder, gentler Goku we see in the series. Goku was raised as a human on Earth, and did not learn of his true nature until his brother Radits came to Earth...
Later, when Goku's rage was awakened by witnessing the death of his friend Kulilin, his Super Saiyajin power was awakened. When he goes Super-Saiyajin his eyes turn green and his hair gold. Currently, Goku can go to the 3rd stage of Super-Saiyajin power.
Gohan is a very good-natured boy and unlike his father, hates fighting, although for a while he was more powerful. He is very polite towards just about everyone except to his enemies, and when he's pissed off. Gohan is very bright and studies very hard, although this is somewhat due to the wishes of his very strict mother...
Gohan is the first character in the series to have gone Super Saiyajin-2, during the Cell Game; the power of his anger was kindled by watching his friends get beaten up, and the death of Artificial Human #16 sent him over the edge. It was roughly at this time that the series switched main characters from Goku to Gohan.
Piccolo is a native of the planet Nameck, although he doesn't learn this for quite some time after his first appearance. As a Nameck, he can regenerate almost any part of his body lost. (usually the arms) He can also extend the range of his arms till they're really long. Later, he goes to Nameck to help Kulilin, Goku, Gohan, and Vegita fight Freezer.
After the Tenkaichi-Budokai, Piccolo becomes more and more of a "good guy." Eventually he becomes Goku's friend; fairly recently he got pissed off at Trunks and Goten for calling Goku weak. Once one of the most powerful characters in the series, he has sort of reached a "middle ground," neither weak nor strong among the Z fighters.
Vegita has a continual rivalry with Goku, or "Kakarott," (Goku's Saiyajin name) and is obsessed with beating him in combat just to show that he is the stronger. This obsession with trying to be stronger than Goku proves to be a fairly large weakness as well.
Despites his efforts to show otherwise, Vegita has changed and has developed a conscience, and deep down cares about his friends. Furthermore, he is also somewhat emotional at heart. He demonstrates this when dealing with his son Trunks, most notably just before he died trying to kill Majin-Buu. Although he pretends to hate Goku with a passion in reality he respects his power and maybe even likes him a bit.
After giving the medicine to Goku, Trunks goes back to his own time, but returns again three years later to help the Z fighters battle the Artificial humans. During this time, he becomes extremely powerful and surpasses every other one of the Z fighters except for Gohan and (maybe) Goku.
The Trunks here, "Future Trunks" and the "current Trunks" are radically different characters. "Future Trunks" is very polite like Gohan and has very strong values, but the current "Chibi Trunks" has a bad attitude and is somewhat like his father. Chibi Trunks never truly reaches the power level of his future counterpart; possibly because he lived through mostly peaceful times, whereas Future Trunks did not.