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 Dragon Ball, 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.
All 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 only surviving Z-Fighters after the Saiyan Saga.
In the Frieza Saga, Piccolo was wished back to life on Namek to fight Frieza. Tien, Chiaotzu, 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 human 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. Goku sacrificed himself to take down Raditz.
Saiyans Saga Part II: Piccolo, Gohan, Krillin, Chao-Zu, Tien, Yamcha, Yajirobe. Later Goku.
The Z fighters met up for a team effort against Nappa and Vegeta. Piccolo, Chiaotzu, Tien, and Yamcha all died. Goku's life was restored.
Freeza Saga: Goku, Krillin, Gohan. Later Vegeta and Piccolo.
The Z fighters embarked on a mission to Namek to use the Namekian Dragon Balls to restore Kami, and through him, the Earth's Dragon Balls. Krillin died and Piccolo was brought back to life. Shortly after returning to Earth, the rest of the Z fighters were resurrected.
Android Saga: Goku (incapacitated), Vegeta, Piccolo, Tien, Krillin, Gohan, Future Trunks
To defend Earth against the androids built by Dr. Gero. In an alternate future, all the members are slaughtered with the exception of Gohan and the young Trunks. Trunks comes back from the future to aide in this battle.
Cell Game: Goku, Gohan, Vegeta, Future Trunks, Android 16, Krillin, Yamcha, Tien
The Z fighters band together to defeat Perfect Cell. Goku, android 16, and Future Trunks die. Trunks is brought back to life shortly after Cell's defeat.
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.
The Z fighters attempt to stop Babidi and Majin Buu. All Z fighters are dead at some point in this arc. Everyone is brought back to life at the conclusion of the fight with Kid Buu.
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)