To my understanding, the Galactic Republic has a group of separatists under Count Dooku, and they later on become the Galactic Empire?
The Confederacy of Independent Systems (CIS), also known as the Separatist Alliance, the Separatist State, or simply as the Separatists, was a galactic government that began as a loose separatist movement which stood opposed to the Galactic Republic. In the wake of the Naboo Crisis, various planetary and sectorial governments, as well as some galactic mega-corporations, declared their independence from the Republic who refused to comply with the excessive taxation and corruption that plagued it, resulting in the secession crisis and a massive military build up in the lead-up to war.
In short, the relatively reasonable goals of the Separatist Alliance (e.g. lower taxes and increased independence from the Galactic Republic) were co-opted by Sidious/Palpatine. He and Dooku ordered two large armies and gave one to each side; a gigantic droid army built on Geonosis for the Seperatists and a clone army bred on Kamino was given to the Republic.
The Jedi were tricked into commanding the Clone Army and at an opportune moment Sidious killed the Jedi (using the clone troops) and the Separatist leadership (using Darth Vader). This caused the Droid armies to shut down and put the Clone Armies under Palpatine's direct control.
With the Jedi dead, the droid armies destroyed and the Separatists decapitated, Palpatine had no-one to oppose him in turning the Republic into The "Empire" that we see in episodes IV, V and VI.
The separatists where used as a pawn by Darth Sidious to gain power.
He systematically used them to gain more and more power throughout the prequels.
He sends the trade federation upon Naboo, where he was a senator. Because of the Chancellor's unwillingness to take swift action, Palpatine convinced Queen Amidala to put in the vote of 'no confidence' against the current chancellor. This resulted in him being stripped of the possistion and Palpatine taking his place (with some Sith tricks).
The machine union was also part of the separatists. They built the droid army, which Chancellor Papatine used as reason for needing a clone army (one that he had already set into motion long before making his power move). The clone army was then used to exterminate the Jedi and also begin the makings of the Galactic Empire's stormtrooper regime.
After Palpatine was in complete control of everything, he had Vader kill all the separatists to clean up the remainder of any evidence of what he had done to gain power, as well as end the perceived conflict in which he used to justify the creation of an Empire.