He is the second son of Chi-Chi and Goku, and brother to Gohan. As a child, Goten looks almost exactly like Goku did when he was young. He is incredibly strong, and turned Super Saiyan for the first time when he was 6, by accident, during a spar with Chi-Chi.
During his childhood he was good friends with Bulma and Vegeta's son Trunks, who he would have "fight games" with. This served to increase their respective power levelss significantly, compared to the other Halfbreeds that we see at that age.
In one of the later Dragon Ball Z movies, Broly attempts to kill Goten due to his similarities to Goku in most aspects (refer to Broly for details).
Goten and Trunks fuse into Gotenks using the fusion technique (as apposed to the "proper" fusion employed by the Elder Kai's earrings), who has the ability to turn Super Saiyan 2 and 3. No other Halfbreed turns Super Saiyan 3, only the fused character. Later in Dragon Ball GT, Goten has turned Super Saiyan 2, and can become very serious during his battles. However he always seems to be slightly behind the one-year-older Trunks. Due to Goten having the genes of Goku, it is speculated that he would eventually become significantly more powerful than Trunks, who has only Vegeta's genes to draw from.