He is simply the most powerful Saiyan to ever live. As only an infant Broly had a power level of over 10,000 - compared to Goku's level of 2. Broly's father Paragus became increasingly worried about Broly's inability to control his rage, so Pargus had a scientist create a necklace capable of restraining Broly. This was attached to him in his sleep for obvious reasons.
Note: Goku and Broly shared a special bond as they were born on the same day.
As an infant Broly became upset at Goku's constant crying (in the next pod), even though he was so much more powerful. This seemed to carry on into adulthood, as Broly's next encounter with Goku lead to him breaking the restraint of the necklace. Future Trunks, Goku, Gohan, Vegeta and Piccolo were unable to match Broly until they channeled their energy into Goku, who used the energy in one last attack, which only temorarily incapacitated Broly.
After the fight with Goku, Broly (somehow) lands on Earth in pursuit of revenge. Broly slipped into a coma in the impact crater, which then filled up with water and froze over. While Broly was in the coma, Gohan grew up, Goten was born and Goku died. As a child, Goten was very similar in appearance to Goku (as a child), they even sound the same. By matter of chance, Goten happened to cry within hearing distance of Broly, which awoke the Saiyan due to the similarities to Goku.
Once awakened, Broly tried to seek his revenge on Goten, mistaking him for Goku. It was only after spiritual help from Goku that Gohan and Goten were able to finally kill Broly. This was achieved with a dual Kamehameha energy attack. It should be noted that this scene in the movie appeared to show Goku assisting in the Kamehameha attack, however he only provided moral support, basically.
This is not the last we see of Broly. In a later movie, it is revealed that a local village Witch Doctor preserved a sample of Broly's blood, which was used to recreate Broly whole, using Bioengineering. Goten and Trunks defeat him.
Broly is otherwise known as The Legendary Super Saiyan.