Yeah, you might as well watch AotC just to get a refresh, but the concise version is:
There's a civil war going on. One side is the Republic that has stood for a thousand years, the other side is the Confederacy of Independent Systems, a collection of planets that have chosen to leave the Republic. The CIS, also called the Separatists or Separatist Alliance, feels that the Republic has grown bloated, corrupt, overly regulatory and cripplingly stagnant and bureaucratic. The Separatists also feel that corporate taxes are too high, and point to the Naboo crisis from ten years prior as proof of all this. In fact, the Trade Federation from the crisis on Naboo is a part of the CIS, as are other such business guilds and interstellar corporations. (History paints both the Trade Federation and Naboo as victims in the conflict - the TF as victims of excessive regulation and taxation, and Naboo as victims of the Republic's inaction to resolve anything; Naboo had to unite the two peoples of their planet and fight off the Trade Federation on their own, according to history.)
Essentially, the Separatist cause is ultra Libertarian. They didn't want to go to war, but merely to declare their independence, at which point the Republic suddenly conjured a "Grand Army" to bully them into staying. They also don't trust the fact that the Republic has a strange, secretive, supernatural cult with super powers on their side. People are familiar with the idea of the Jedi, but not everyone really knows their details or particularly trusts them. The Republic suddenly making them the generals commanding the new Clone Army wasn't exactly a great PR move.
So the Separatists went to war, and while they might not have a "Grand Clone Army" led by weird wizards, they DO have industry. The droid factories on Geonosis and other industry driven Separatist ally planets can turn out "clankers" that may not be as strong battle units as living clone troopers, but can be produced faster and in greater numbers.
As far as anyone knows, the CIS is led by Count Dooku, its founder, president and military commander, who has a leadership council serving with him, as well as a collection of strong military Generals.
(Of course, what the people of the Galaxy DON'T know, is that Dooku actually has a boss too, and that said boss is also the leader of the Republic as well, and the whole thing was engineered for profit and to change the Republic's leadership structure. Nor that Dooku's boss/the mastermind behind it all even engineered the Crisis on Naboo ten years earlier just to set the stage.)
...I guess that wasn't too concise after all.