At the beginning of DBZ, we're introduced to a 4-year-old
half Saiya-jin named Gohan. Son to our hero, the Saiya-jin
Goku, and the human Chi-Chi, he's a hybrid with incredible
fighting potency. Saiya-jin kids are usually born with more
fighting potential (which means very little since all the
Saiya-jins in the show constantly out does themselves).
In the beginning of Z he is kidnapped by his uncle Raditz
and trapped inside of his Saiya-jin Space Pod. He hears his
dad's cries of pain and his rage consumes him. His power level
jumps up to over 1,300 (note that Goku's power level is around
300-400 at this point) and he breaks out of the space pod,
then delivers his uncle, Raditz, an armor-shattering headbutt.
Later, he's taken by Piccolo to be trained so he can fight against the elite Saiya-jin, Nappa and Vegeta. He's forced to spend six months alone in the desert for survival training, and the next six training under Piccolo. When the Saiya-jin arrive, he helps to defeat Vegeta by going Oozaru.
On Namek, he helps a lot in gathering the Dragon Balls, and forms great friendships with both Krillin and Dende. Once again his potential is seen when Frieza almost kills Krillin, and "the only way to help him was through Frieza."
After finding out about the Androids, he trains under Piccolo and Goku for three years. He later trains in the Room of Spirit and Time under his father; this is when he goes Super Saiya-jin.
During the fight with Cell, he goes Super Saiya-jin Level
2 (note that these levels were described in numbers only after
Goku goes SSJ 3) when he sees his friends getting destroyed
by the Cell Jr.'s. He takes care of them quickly, killing
them all with ease. He's responsible for his father's death
-- in a way -- because he could've easily killed Cell, but
just toyed with him, and Cell ended up killing Goku. Gohan
eventually vaporizes Cell with a one-armed Kamehameha and
a little help from the rest of the Z warriors as well as Goku
He spends the next seven years hitting the books, and then
starts high school, where he meets his future wife, Videl.
He also becomes the Great Saiyaman, which is what he enters
the 25th Tenkaichi Budoukai as. The Majins drain him of his
energy, and he's escorted by Kaioshin to the holy planet,
Kaioshin Kai. This is where he get's to a new level (not a
SSJ level mind you). After his training and the bringing out
of more power by the older Kaioshin he get's strong enough
to beat the current Majin Buu (right after splitting with
Fat Buu). Buu then get's smarter and ends up absorbing Super
Gotenks and Piccolo (for brains). At this point he's far weaker
then Majin Buu and simply can't do much in the battle field.
Soon he'll get absorbed by Buu too....
Later on in GT he becomes a total nerd/scholar and does very
*Note that the word Kaioshin will not be used in the FUNimation
dub or at least for the younger Kaioshin seen in the show
as he is called the Supreme Kai.