There were actually two brothers, Tiberius and Gaius Gracchus, known as the "Gracchi". They were demagogues who promoted the interests of plebians and socii in Rome. Their movement signalled the downfall of the Roman Republic. When they could not overthrow the privileges of the long-born Romans (known as Optimates), they started efforts to overthrow the republic by force. These efforts ended in disaster and the traditional forces in the city had them and their followers murdered. This set a bad precedent in which the fate of Rome would be decided by violence and power, not by the votes of the original citizens, the Optimates.
The Gracchi were followed by Gaius Marius, who was a novus homo. He saw the power the disenfranchised could give him. Marius, repeatedly elected Consul, did everything he could to become a dictator and essentially restore Rome to a kingship--something it had foresworn centuries ago when the republic was founded. When the champion of the republic, Sulla, left to fight Mithridates, who led a rebellion of the Greeks against Roman rule, Marius took advantage of the opportunity to dissolve the senate, and make himself dictator by force and started executing republicans.
At this, Sulla returned to Rome with his legions, instituting a civil war. Marius responded by offering freedom to any slave would fight with him against Sulla. Originally, only full citizens who owned land were allowed to be in the army. For Marius to arm slaves was a complete abandonment of the principles of the republic and an attempt at despotism. Sulla entered Rome with his legions--an illegal act--and eventually defeated Marius the Younger (who committed suicide) and killed thousands of his supporters (the elder Marius had died earlier). All of the slave-soldiers and other usurpers were hunted down and put to death. Then Sulla restored the laws and institutions of the republic and retired.
This peace only lasted a short time. Julius Caesar, a follower of Marius with family ties to the Marians, followed the exact path that Marius had--gathering power from the plebs and non-citizens until he had enough power to make himself dictator. This was the beginning of the empire and the final end of the Roman republic.
(There is a story that Caesar was originally on the lists to be executed, even though he was a young man. Powerful and connected women in the city interceded to protect Caesar and Sulla's men came to him and insisted Caesar be taken off the lists. Sulla warned them, "In this young man, there are many Mariuses." Nevertheless, Caesar was taken off the lists.)