The Republic, as it was commonly known, had a highly democratic government that was able to sustain itself for over twenty-five thousand years (according to the Expanded Universe).
The main organ of the Republic was a Galactic Senate which featured several appointed senators from each member planet. The senators would then elect a Supreme Chancellor from amongst their ranks who would serve as the Republic's head of government (there seems to be no Head of State) and chief diplomat. Based on the films, we can see that the Senate follows pseudo-parliamentary rules. In Episode One Supreme Chancellor Valorum is forced out of office by the Senate in a Motion of No Confidence.
Regional government in the Republic was in the hands of elected monarchs for each planet, who had the power to appoint Senators to the Senate. There were also small elected assemblies to assist each monarch. The monarchs apparently also had the power to vote in the Senate and introduce legislation. It was Queen Amidala of Naboo in Epsiode One who introduced the non-confidence motion mentioned above.
The Republic's greatest failing was its lack of formal organization. Much of its operation was based on non-binding constitutional conventions, which were often ignored and undermined by corrupt and power-hungry politicians.
The Republic was made up of about a million worlds. The Clone Wars, which occured from 22 BBY to circa 20 BBY, was the first full scale war since the Republic formation(though there were other wars throught the Republic's history such as the Hundred Years Darkness(7,000 BBY), The Great Hyperspace War(5,000 BBY) and The Great Sith War(4,000 BBY).
The Clone Wars occurred when several of the Republic's member Star Systems and organizations united in order to separate from the Republic. This unified organization became known as the Confederacy of Independent Systems. Tensions between the Republic and the Separatists eventually broke, and the Clone Wars began as chronicled in Attack of the Clones.
By the time of The Phantom Menace, the Republic was incredibly corrupt. Senator Palpatine of Naboo became the Supreme Chancellor in 32 BBY. Traditionally, the Chancellor could only have two 4-year terms, but Palpatine stayed in office much longer. He was granted Emergency Power in a motion led by Senator Binks to deal with the Separatist Crisis. Eventually he declared himself Emperor of the galaxy, and the Republic ceased to exist.
Decades later, shortly before the Battle of Yavin, Palpatine dispersed the Imperial Senate and the last remnants of the Galactic Republic were swept away.
After the Galactic Civil War, the Galactic Republic was re-established as the New Republic.