The Galactic Republic was a galaxy-spanning republic that existed for a thousand generations before it was dissolved by Chancellor Palpatine and replaced by the Galactic Empire. In the Imperial era, it was referred to as the Old Republic.
The Post-Russan Republic
Following the Battle of Ruusan, the Galactic Republic entered its last 1000 year golden age. Enjoying an age of prosperity unmatched in its history, the Republic dissolved almost all of its military forces and brought the Jedi under the official auspices of the government. The post-Ruusan Republic was centered around its legislature, the Senate. Representation on the senate was awarded on a per sector basis, with sectors being grouped together as the number of sectors increased, so as to keep the actual number of Senators around 1000. Senators controlled the executive branch, by electing one of their own as Chancellor and by being able to remove the Chancellor with a 'Vote of No Confidence.'
The Decline of the Republic
Strong enough to resist any outside assault and lacking utterly in any enemies, the Republic waned from the inside. Its modest starfleet was only capable of the most modest interventions in its own members affairs and as a result brushfire wars were fairly common. Ironically it was the demilitarized Jedi who provided the best defense against the unraveling of civilization. As time wore on, corruption became more and more of a problem in the upper halls of the Republic. At the same time, massive corporations like the Trade Federation slowly crushed many of the independent economies of the Outer Rim, depressing the development of the least populated and poorest areas.  Not unsurprisingly, the other regions of the galaxy failed to notice when the Outer Rim slowly slid from the control of the Republic.