The Separatists are the Planets and races that band together that no longer want to be part of the Old Republic. Whilst the main leader is identified as being Count Dooku, the organisation is being controlled by a Sith Lord at the heard of the Old Republic. The Separatists included the Trade Federation that was duped into a trade war with Naboo over trade routes. It was the siege of Naboo that started the break aways.
It was the Separatists that designed and built the first Death Star as shown in a holographic display in Star Wars II - Attack of the Clones. The Death Star duly passed into the hands of the Galactic Empire.
In Star Wars III - Revenge of the Sith, the Separatists are led by a new leader, General Grevious, a cyborg. After Obi-Wan Kenobi has killed Grevious, Palpatine orders Order 66 which is for the Stormtrooper to turn on the Jedi and kill them. Palpatine tells the Galactic Senate that Jedis are being hunted down as they are traitors.
After Palaptine has been saved from arrest from Mace Windu, Palpatine anoints Anakin Skywalker as Darth Vader and instructs his apprentice to go to the lava planet of Mustafar to kill all the remaining Separatist leaders include Gute Nunray.