Ah, Trunks. Son of Vegeta and...Bulma o_0.To make a long story incredibly short, thought it still is V. long, after Goku defeats Freiza on Namek, both Goku and Freiza survive and head to Earth, Goku is supposed to kill Freiza once they both arrive and then later die of a heart virus.
Some years later, two androids are supposed to appear and kill every character in DBZ except Gohan, Trunks, Bulma, Master Roshi, Puar, and Oolong. Piccolo was the first to try and stop the androids, and the first to die, upon his death Gohan became a Super-Saiyan.
Some years afterwards Gohan and Trunks try to stop the androids and nearly get killed and Gohan loses an arm. Trunks asks that Gohan train him to become a Super-Saiyan like him. When the androids attack another city Gohan and Trunks go after them, but before they do, Gohan knocks Trunks out and says something really cheesy like "Stay alive so you can defeat the androids one day, I know you can." even though Trunks can't hear him. So Gohan goes and tries to destroy the androids and gets killed. When Trunks wakes up, he goes to the city where the androids were attacking and finds Gohan dead on the floor, he refuses to believe Gohan is dead and keeps trying to wake him up. He then says something really cheesy like, "You were everything to me!" and he shakes his head and starts screaming and becomes a Super-Saiyan.
He then spends the next 5 years training to defeat the androids, and when he hears that they are attacking again he tells his mom, Bulma, that he's going to go stop them. Bulma then says that he isn't any stronger than Gohan was when he died. Trunks fights the androids and soon runs out of energy and finds out the androids never run out of energy and gets beat up REALLY bad and nearly gets killed. When he wakes up again, he's in his house and Bulma is at his side. He then gets medicine to treat the heart virus Goku had and time travels to the time when Freiza arrives on Earth (Freiza arrives less than an hour before Goku does) and Trunks kills Freiza's henchmen, Freiza, and his dad, King Cold.
When Goku arrives, Trunks and Goku fight for fun (Trunks says its to make sure it is Goku, though Goku is the only Super-Saiyan from that time period...). Trunks then explains everything to Goku about his heart virus and the androids, he even tells them the exact place, date, and time the androids will show up, and returns to his own time.
The Z warriors then train for about 5 years until the androids show up, and Trunks returns to their time to check up on things and finds that the androids that showed up in this time aren't the same androids from his time and that one of the androids was already destroyed by Vegeta (he's now a Super-Saiyan).
So he flys torward the battle field and tells everyone the androids aren't the same and also finds that the heart virus has started attacking Goku's heart. Meanwhile, the other android they were fighting takes off torward his laboratory to activate androids 17 and 18 (the ones from Trunk's time).
Vegeta, now thinking he's so strong, wants to wait till 17 and 18 are activated to destroy them. Piccolo and Trunks think its best to hurry and destroy them before they are activated and Trunks tells Vegeta that he has no idea how powerful these androids are. Then Vegeta punches Trunks in the stomach (Vegeta is a Super-Saiyan at the time and Trunks is not), and goes after Dr. Gero's creations.
When they arrive at the cave, android 17 destroys the remote control that Dr. Gero used to deactivate the androids last time and also destroys Dr. Gero.
So then the androids and the Z warriors start to fight but only one at a time first Vegeta vs. 18 and the others are not supposed to interfere. Pretty soon Vegeta gets beat up and is really injured and Trunks tries to save him only to end up getting beat up by 17, then everyone else jumps in except for Krillin and get beat up and the androids leave but 18 kisses Krillin on the cheek first.