This is a tough one to answer shortly or definitively. I'll try to hit some key events and themes, at least until a Roman specialist can answer.
In the Republic, the high office that governed Rome was the consulship. Two consuls were elected for a term of one year. The idea was that one man alone shouldn't be trusted with the power. The consulship came to be seen as the head of the "cursus honorum" or "race of honors" of Roman magistracies. An ambitious patrician might serve in the military, be elected to the quaestorship (work in financial administration of Rome or be effectively a vice-governor to a province), the aedileship (putting on public events and managing public resources), the praetorship (becoming a judge), and finally the consulship. After a consul retired from his year of service in the late Republic, he would typically be named a proconsul (military governor) of a Roman province. A good proconsulship was a rich prize, as it offered the opportunity for conquest and taxation.
After a class struggle in the early Republic, plebians could pursue a similar course, or could be elected to the tribunate. The tribune could propose laws through the popular assembly and could veto anything to come out of the Senate. For some, the tribunate was a terminal office, for others, it was a stepping stone to the consulship. The office was created as a part of the package of rights and privileges extended to the plebians following the several Plebian Retreats (combination labor strike and threat to found another city) in the early Republic. Originally, the division between patricians (those of aristocratic families) and plebians (those not) was great, but by the late Republic, there were many rich and politically powerful plebians, some patrician houses had been virtually impoverished, rich plebians could marry up, and in the case of Clodius, a patrician even sought to be declared a plebian in order to run for the tribunate and use its powers against his enemies. Your average patrician was certainly above your average plebian, but the line was no longer absolute.
Much of the politics of the later Republic are defined by conflicts between the optimates who championed traditional conservative Roman values, and the populares, populists who gained power by promising reform for the masses. Both were strategies for gaining power, and several men danced the line between the two.
Tradition has it that Rome was founded circa 753 BC, and ruled by seven kings. After the fall of the final king, Tarquinius Superbus (expelled by a man named Brutus) in 509 BC, the Republic was founded and Romans became violently opposed to the idea of monarchy. The title rex ("king") was hated and sometimes hurled at an ambitious man by his enemies. A king was a tyrant of the past, or a mark of uncivilized barbarians.
It's hard to know when to start when talking about the fall of the Republic. I'd start with the tribunates of two brothers, Tiberius and Gaius Gracchus. They were born plebians, but within a wealthy extended family that contained both plebian and patrician branches.
Tiberius Gracchus became tribune in 133 BC, pursuing radical land reform. Reportedly, while travelling through the Italian countryside, he was dismayed to see the land once worked by the agrarian Roman ideal of the yeoman farmer transformed into wide estates owned by wealthy Senators and worked by gangs of slaves. Traditionally, a Roman man had to own land in order to serve in the army, and so Tiberius saw the concentration of land ownership in the hands of a few as a weakening of Rome, both morally and quite literally. He faced massive resistance from other magistrates (including other tribunes - there were more than one) and the Senate. He campaigned (unconstitutionally) for reelection, as a man holding a magistracy could not be prosecuted, and his enemies had promised to prosecute him as soon as he left office. When he was warned of an assassination attempt in public, he attempted to shout over a crowd and motioned to his head (to say, in effect, that they were coming for his head). His enemies seized on the confusion and claimed he was saying that he wanted a crown on his head (raising the spectre of the hated kings). He was killed along with hundreds of his supporters, in the first open political bloodshed of the Republic. Ludicrously, his enemies still tried to honor the law against bloodshed in the city by beating them to death with sticks and stones, instead of killing them with blades.
A decade later, in 123 BC, his brother Gaius was elected to the tribunate, and much the same story played out. He supported land reform and other benefits for the poor, along with advancement for several sectors of society, again campaigned for re-election to protect himself and keep pushing his reforms, and as he was about to be assassinated, he committed suicide. The Gracchi supporters were slaughtered by the thousand. Republican politics had been bathed in blood, but not for the last time.
From here, I would jump ahead to the rivalry between Gaius Marius and Lucius Cornelius Sulla, two great generals. The story of Marius and Sulla is far too long to recount here in detail, but here are some incredibly abridged notes. Marius, a great general, responded to the threat to the military reserve seen by Tiberius Gracchus, reforming the military (allowing the landless to join) and cultivating an intense loyalty among his soldiers. He was elected to the consulship a record seven times (though not all in a row).
Sulla was a younger man, a patrician of a poorer family who enjoyed a debaucherous youth in the friendship of prostitutes, actors, and other ne'er-do-wells, and came to prominence while serving under Marius in a war in Numidia, when Sulla captured the enemy king. Sulla followed Marius to other military victories, including against invading Germanic tribes.
In his sixth consulship, Marius attempted to push through a land reform package awarding farmland to his veterans, but faced heavy opposition. Following more violence and the end of his term, he receded from public life for a time, and indeed was rather old at this point. While the two were both somewhat involved, it was Sulla who largely put down a revolt by non-Roman Italians in the years that followed. Sulla was elected consul and the Senate awarded him a rich eastern command (rather than giving it to Marius) in a war against the King of Pontus, Mithridates, who had invaded Rome's eastern provinces and ordered the slaughtering of Italians in Greece. The one who stopped Mithridates would win great honor, with the added benefit of gathering riches in the east. When Sulla was delayed with his army in Italy (uppity non-Roman Italians, again), a tribune allied with Marius managed to call a plebian assembly to overturn the Senate's decision and award Sulla's command to Marius.
In 88 BC, Sulla, the one physically if not legally in possession of the army, turned it to march on Rome, an unprecedented act by a Roman general. Marius organized a desperate resistance, but it was smashed by Sulla's forces, and Marius fled the city, escaping capture and a death sentence. Sulla installed supporters in key areas, but despite his attempts to sway the elections, a Marian was elected to one of the posts. Sulla forced the man, under threat of violence, to swear to uphold Sulla's imposed order, but that oath didn't hold long. Sulla departed for the east to face Mithridates. In the mean time, Marius, who was busy raising an army in the African provinces, returned to seize the city himself, slaughter Sulla's supporters, and take his seventh consulship. He died shortly thereafter, in 86 BC, with Sulla still in the east.
Sulla reclaimed the conquered provinces, but in his need to race back to Rome to deal with the Marians, reached a peace with Mithridates short of the complete destruction of the man that so many Romans thirsted for. Mithridates would return as a major threat to Roman interests again in the future. Sulla returned to Italy, marched on Rome again, had himself declared dictator (an emergency magistracy in which the Senate could empower one man to hold executive power). During this time, he reordered Roman law, set new rules for the magistracies (including barring tribunes from becoming consuls), and engaged in mass slaughter of first the Marians and then anyone who ran afoul of - or had assets coveted by - his allies, via regular proscription lists. After his dictatorship, Sulla declared the Republic "restored" from the Marian overreaches, was elected twice to the consulship, retired the countryside, and died in 78 BC. Even after Sulla retired and Roman elections began "as normal," his supporters remained a significant power bloc in Rome, occasionally calling him to intervene during his remaining years.
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