Son Goku is the primary hero of the Dragon Ball series. The series started out with Goku as a child of age 12. Once he grew up and his son was born, the TV series' name changed from Dragonball to Dragonball Z. At the end of the series, Goku was roughly 45 years old.
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, including Ultimate Super Saiyan.
Son Gohan is Goku's first son. At the beginning of the series, Goku and Piccolo team up in an effort to save Gohan from the Saiyan Radditz. The three battle it out in a frenzy. The advantage that Raddiz has over Goku and Piccolo, however, is too much for the both of them. Eventually Raddiz beats and tortures Goku. Gohan hearing his father's cries, gets enraged and shows his true powers. This is where we see Gohan's powers for the first time. At the end of the series he was 26 years old. For a short while after the second time Goku died in the series, Gohan became the primary character. Gohan is the most powerful non-fused character in the DBZ world, but lacks the love of fighting his father has and thus rarely uses his full strength. 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.
Vegita is the prince of all Saiyans (Saiya-jins). He was born an elite Saiyan and has the ability to become a Super Saiyan. Along with Nappa, Raddiz and Goku (Kakarot) he is one of the four survivors of his home planet "Vegita"*. The Saiyans were part of Freeza's organization and used as tools to take over and prepare planets for sale. Freeza took Vegita as a child and sent him out on special missions. While out fulfilling one of those missions, his home world was abruptly destroyed (this happened when he was a child). As Freeza's henchmen, Vegita was regarded as one of the more powerful soldiers in the galaxy. He always knew that Freeza destroyed planet Vegita and plotted to one day overthrow the vile ruler. At birth Vegita was calculated to have the highest class as a Saiyan. That perception changed when he met one warrior he could not defeat: Son Goku. As a man with great pride in being the best Saiyan he couldn't accept the fact that Goku, a low class Saiyan, was always one step ahead of him. He always try to best him in combat and this has been his obsession ever since. With his pride and huge ego he is always trying to show that he is the most powerful in the universe. People are always saying that Vegeta was too weak to ascend to Ultimate Super Saiyan level in the Hyperbolic Time Chamber the way Goku and Trunks did, that is a lie. As a matter of fact, Vegeta is the first person that shows the Ultimate Super Saiyan power. However, he showed this power only once because he that while being in that form, his speed would greatly decrease. He showed his power the first time he fought Cell. After he was finished powering up into Ultimate Super Saiyan, He took a breath to let his muscles a little smaller so he was able to have some speed when he was fighting. If you are one of those people that still believe that Vegeta never reached the level of Ultimate Super Saiyan then I have more proof for you. The first is that the only reason Trunks was still stronger was because he trained harder in the Hyperbolic Time Chamber. The second proof is that Trunks himself says that he reached it in these exact words "He's ascended" and Cell also says it when he explains to Trunks that he was slow because of his bulky muscles. If you still don't believe me then watch the 3 episodes after "17's End".
Piccolo is the reborn Devil himself. Many years ago, A Namekian was sent to earth in fear that a comet would soon hit the planet Namek. The man we know as Kami-sama (which translates into god or the god of earth) wanted to take the place of the watcher of earth. But in order for him to do that he had to purge the small amount of evil that was within him. When Kami-sama did this his spirit split into two. Thus, from the expulsion of Kami's evil side, was the birth of an evil entity called Piccolo-Diamao (Piccolo the demon). Piccolo-Diamao was evil and destructive causing cities to disappear. Eventually this evil being came across Goku (when he was young). They fought an epic battle, with Goku killing Piccolo-Diamao. On his dying breath he spat out an egg which hatch into the Piccolo in the Z team. Young Piccolo was evil like his father. A native of the planet Namek, he has the ability to regrow parts of his body that he lost. He and Goku were enemies as teenagers and fought many times. However when he was forced to train Goku's son Gohan before Vegita and Nappa were to come to earth, a different side of Piccolo is revealed and we see that he is not entirely evil after all. Gohan's innocence and friendliness help Piccolo become a different person changing him to be on the side of good instead of evil. Further in the story Piccolo develops a paternal affection for Gohan, protecting him. During the Android saga, Kami-sama nulls on the decision to remerge with his "evil" counter part noting that Piccolo has good in his heart. Kami eventually decides to do so forever putting him on the side of good and fighting with the Z team.
In the alternate timeline, all of the Z team members were killed because of two destructive androids, except for Goku who never gets to fight because of a heart infection. The only survivors were Son Gohan and Trunks. Trunks is the son of Vegita and Bulma. Gohan trained him to become stronger and to reach the level of Super Saiyan. One day the androids appeared and, Gohan confident that he can beat them, had enough of the destruction. By protecting the inexperienced Trunks, Gohan knocked him unconscious and fought them alone. Gohan is powerful, but is killed nevertheless. Trunks grieves deeply and vows to one day avenge Gohan's death. Meanwhile Bulma builds a time machine for Trunks so he could go back in the past to find Son Goku. She believes that if he were alive, things would have been different. Trunks comes to the past twenty years from the future in a effort to change history. Much like his father, he is hard and aggressive. But unlike his father he is very polite and has values. He thinks in conventional ways and is very civilized when making decisions. While Trunks was young, he grew up in awe of the father he never knew. Trunks was always anxious about meeting his father. He was both impressed and disappointed once they did. During the battle of Cell however, Vegeta showed emotion upon seeing the death of his son (that may be subject to debate for all you hard core nuts!!) Vegita then screamed with passion and attacked Cell. The Z team revived him after Cell was defeated. He decided to return to his timeline once the danger was over. Trunks had trained in that time so much that his power surpassed the two androids in his time. When he returned to the future, he not only took with him the skills to defeat the androids but also the knowledge of the father he loved so much.
The first child of Vegita and Bulma. Young (Chibi) Trunks is very different from future (Mira) Trunks. He is a little mischievous, and has a bad attitude much like his father. Although he means well most of the time, he always gets himself and Goten in trouble. Chibi Trunks never reaches his potential because of the fact that he lives in times of peace.