He is the Legendary Super Saiyan that is born every thousand years on Planet Vegeta. He was born with a Power Level of 10,000. In his infancy, Son Goku's constant crying infuriated Broly and eventually drove him to insanity. Broly was sentenced to die by King Vegeta but before he could be executed, Frieza destroyed the planet. Broly's power helped him and his father, Paragus, survive.
Broly's power was so immeasurable, that in the first movie, Son Goku was only able to defeat him by having the rest of the Z-Fighters transfer their power to him. Though Son Goku defeated him, Broly survived and crash landed on Earth being frozen in a lake.
He was reawakened from his slumber when Son Goten's crying made him think of Son Goku and he engaged Son Gohan, Son Goten, and Trunks in battle. He planned to destroy the entire Earth with only Son Gohan and Son Goten holding his energy back. The Dragon Balls then summoned Son Goku to Earth, and with some help from an injured Trunks, overpowered Broly and launched him into the sun, finally killing him.
However, a mad scientist clones Broly from some blood by orders of Mr. Satan's rival Lord Jaguar. Containing Broly's hatred, the clone breaks out of stasis before being complete and becomes corrosive. The clone is destroyed when it falls in water and hardened, allowing it to be destroyed, effectively destroying Broly for good.