The Galactic Republic
The politics of the series are vaguely Roman. The Republic, complete with senators, is in danger of being made over into a dictatorship by a powerful man, Palpatine, who wants to make himself emperor. The actual, physical Senate is a giant room filled with open-air flying saucers, where aliens from all over the galaxy yell at each other. Every world in the Republic sends a representative or five. Queen Amidala was a senator from the planet Naboo, as was the computer-generated shucker-and-jiver Jar Jar Binks. How do you get to be a senator? Nobody knows. Probably something to do with electronic voting.
Jar Jar Binks
Ahmed Best (voice). A member of an amphibious race that also lives on Naboo. Jar Jar gets caught up in the conflict in Episode I. His lame comedy bits and weirdly offensive personality make him the focus of fan disillusionment. All I can say is: He has an accent.
These mystical samurai cops of the Republic keep the peace and do Senate wet work, like when Obi-Wan Kenobi and Qui-Gon Jinn go to Naboo in Episode I to fight the Trade Federation, or negotiate with it, or whatever it is they're doing. Jedi wield the Force, and they're administered by Masters who sit in chairs (appropriately shaped for alien body types) in the penthouse of the Jedi headquarters on the city-planet Coruscant. Yoda is a Jedi Master, as is Mace Windu, played by Samuel L. Jackson, who swears his character will die like a man in Episode III. Jedi are supposed to be as free of desire and emotion as a Vulcan Buddhist. That's why Anakin's love of Padmé is not only painful to watch on screen, but also catastrophic.
The revenge-takers in the eponymous Episode III are the evil counterparts of the Jedi, counting among their members Darth Vader, Darth Sidious, and Darth Maul (he of the double-bladed lightsaber in Episode I). They can often throw lightning from their fingertips, which makes any one of them a Darth and stormy knight. Sith always come in pairs, and they draw their power from the dark side of the Force, a wellspring of anger and hatred—aimed especially at the Jedi.
Ian McDiarmid. Palpatine is secretly Darth Sidious, who becomes the emperor. Hel-lo? Accent! As Sidious, he seems to be the prime mover behind the creation of a clone army. Apparently the idea is to spark galactic chaos so that he can become chancellor of the Senate, and eventually take over the whole show. Palpatine also spends inordinate amounts of time turning Anakin Skywalker from a petulant kid into Darth Vader.
Frank Oz (voice). An extraordinarily powerful Jedi Master, originally played by a muppet in Episode V, Yoda packs a lot of Force into a teeny green body. In the prequels, he worries about the fall of the Republic before anyone else. After the Jedi pogrom, he hides out on the planet Dagobah, where he eventually trains Luke Skywalker to be a Jedi. He has difficulty with grammar, but no accent.
Trade Federation and Separatists
These are the bad guys from episodes I and II. The offensive Asian stereotype green dudes attacking Naboo? Trade Federation. The weird insect guys who set up the gladiator scene and the assembly line at the end of Episode II? Separatists. Count Dooku, Christopher Lee's evil Jedi whose lightsaber curves to the left? Separatist. They're the ones with the droid army.
It might seem, on first consideration, that a weapon of limited range (about 3 feet) that burns through anything it touches (including the wielder) and cauterizes every wound it makes (which definitely limits stopping power) would not be ideal. But Jedi love lightsabers nonetheless.
Advanced lesson: Possibly you get a new lightsaber when you switch sides. Jedi have blue blades; Sith blades are red. Luke's first lightsaber, which Obi Wan told him belonged to his father—Anakin/Vader—was sort of white. It remains to be seen whether Anakin really wanted Luke to have it or if that was another Obi-Wan fib. Oh, and Mace Windu's blade is purple. Sam Jackson makes purple look cool.
What is Jimmy Smits doing in a Star Wars movie? He's the senator from Alderaan, fighting to defend the Republic. Princess Leia's last name is Organa, too—when Queen Padmé gives birth to her twins, Luke and Leia, the girl goes off with Bail to live on Alderaan. That planet, by Leia's account in Episode IV a "peaceful world" with no weapons, meets a bad end as the first after-market test group of the Death Star.
Many, many guys in useless white armor with famously bad aim. In Episode IV they're just average foot soldiers, and not above small talk. One even hits his head on a doorway. But in Episode II, we learn that they're all clones, built to defend the Republic against an equally large and equally computer-generated army of droids. Of course, both the droid army and the clones are actually pawns in Palpatine's grand scheme for domination of the galaxy.