The Old Republic is also known as the Galactic Republic. Its referred to old because it has lasted for thousands and thousands of years. Before Vice Chancellor Palpatine turned the galaxy into his own Empire, the galaxy was controlled by the Senate. The Senate consisted of numerous worlds who would debate in the Senate Building on Coruscant. Palpatine used and corrupted the Trade Federation to bolster his powers and move up the power hierarchy. The Senate had maintained the peace for the thousands of years until Palpatine managed to corrupt and convert it.
Palpatine took control of the Senate when there was a vote of no confidence in Chancellor Valorum and so was duly elected by the Senate to take charge. Its all part of the master plan to control everything. The Old Republic is different to the Jedi Council as the Council enforces peace whereas the Senate makes laws and settles disputes.
A person claiming to be Sifa Dyas orders the creation of a clone stormtrooper arm to be built on the planet of Kamino. However, what they Kaminons don't know is that Sifa Dyas died many years ago so it couldn't have been him who actually placed the order.
In a way, the Old Republic is like the European Union where each country is equivalent to a planet. Each have their own laws and customs and come together for more laws. The difference however is that there is no European Army (Yet). Although there is an European Vice Chancellor or President, it is nothing more than a figure head, he can't order armies into battle.