The Z-Fighters are Earth's first defense against evil. Many of the Z-Fighters were once enemies or rivals of Goku (Piccolo was one of Goku's greatest opponents in Dragonball, Yamcha and Krillin were his rivals, etc.).
The Z Fighters are not officially organized in any way, because it is simply a group of super fighters who share the same interest of protecting their home. However, they all stand together whenever possible to fight off enemies. Goku is the de facto leader of the group due to his incredible success rates. Hercule also calls them "The Golden Warriors" for the hair color of their Super Saiyans members.
ll of the Z-Fighters were present when Raditz, and afterwards, Nappa and Vegeta came to Earth, and many of them died protecting it. Krillin, Yajirobe, Gohan and Goku were the last Z-Fighters after the Saiyan Saga.
In the Frieza Saga, Piccolo was wished back to life on Namek to fight Frieza. Tien and Yamcha decided to stay with King Kai and train. Krillin was killed by Frieza, triggering Goku's Super Saiyan transformation.
After the Frieza Saga, the Z-Fighters were proven not as capable as the Saiyans and became minor characters, giving advice and training the more powerful characters from time to time. Krillin and Piccolo were the only non-Saiyan Z-Fighters to still be two of the major characters.
Pre Dragon Ball Z: Goku, Krillin, Yamcha, Tien, Chao-Zu, Yajirobe, Master Roshi
Formed before the attacks on Earth with the purpose of collect the Dragon Balls and fight King Piccolo.
Saiyans Saga Part I: Goku, Piccolo
Considered the first true incarnation, an alliance formed against Raditz's attack.
Saiyans Saga Part II: Piccolo, Gohan, Krillin, Chao-Zu, Tien, Yamcha, Yajirobe. Later Goku.
Against Nappa and Vegeta
Freeza Saga: Goku, Krillin, Gohan. Later Vegeta and Piccolo.
In a mission to Namek to obtain the Dragon Balls.
Android Saga: Goku (incapacitated), Vegeta, Piccolo, Tien, Krillin, Gohan, Future Trunks
To defend Earth against Dr. Gero androids. In an alternate future, all the members are slaughtered with the exception of Gohan and the young Trunks.
Cell Game: Goku, Gohan, Vegeta, Future Trunks, Android 16, Krillin, Yamcha, Tien
To defeat Perfect Cell.
Post Cell: Gohan, Vegeta, Piccolo, Goten, Trunks
After the death of Goku and the retirement of most of his humans members.
Majin Buu Saga: Goku, Vegeta, Gohan, Gotenks, Piccolo, Vegetto. Tien and Krillin briefly.
Against Babidi and Majin Buu. Later only Goku and Vegeta stands.
Dragon Ball GT: Goku, Trunks, Pan
In a space quest to obtain the Black Dragon Balls.
Freeza Saga: Nail, Dende
Post-Cell Game: Android 18
Majin Buu Saga: Videl, The Supreme Kai, Elder Kai.
Dragon Ball GT: Uub, Majin Buu (good)