The Krogan Rebellions is the name given to the three hundred year period of excessive expansion and growth of the Krogan race. In this time, the Krogan began to quickly expand and colonize other words by forcing the existing races out of them. Due to their incredible strength and fighting ability, they were able to do so rather easily. When the Krogan was asked to stop by the Citadel Council, they refused. Because of their refusal, a war broke out. Even though the Krogans were technically defeated by the Turians, they never really surrendered. The Turians decided to turn to the Salarians, who had developed a biological weapon called the genophage that, when deployed, would cause only 1 out of a 1000 Krogan births to be successful. The Salarians didn't want to use the weapon until necessary, but when the Turians got their hands on it they deployed it instantly.
Mass Effect 1
During the assault on Saren Arterius' base on Virmire, it is discovered that the rogue Spectre managed to develop a cure for the genophage and that he was going to use it to create a Krogan army. The Salarians, with the help of Commander Shepard and his crew, destroy the base and the cure at the same time. When Urdnot Wrex, the Krogan in Shepard's crew, hears about their plan, he snaps. According to him, the Krogan were thanked for destroying the Rachni by being neutered. He sees the cure as an opportunity to rebuild his dying race. Commander Shepard can either convince Wrex to back down and accept it, or kill him.
Mass Effect 2
In Mass Effect 2, Commander Shepard recruits a Salarian scientist by the name or Mordin Solus. During Mordin's loyalty mission, they track down a former apprentice of his who they believe was captured by mercenaries. Upon his discovery, they learn that he was working on a cure for the genophage and wasn't actually kidnapped. Because the scientist was using live test subjects for his experiments, Mordin kills him unless stopped by Shepard (using the Paragon interrupt). The player can then decide whether to keep the genophage cure research or destroy it.
During the investigation leading up to his discovery, Mordin reveals that the modified version of the genophage he helped work on was deployed because they predicted using simulations and statistics that a war would break out resulting in the genocide of the Krogan. The modified version was designed to keep the Krogan numbers low but high enough for the species to survive.