Anakin decapitating Count Dooku in order of Palpatine.
As far as the dark parts of John Williams' score go, almost nothing is as disturbing and unsettling as the track "Palpatine's Teachings". It is one of those kinds of music that can make anything seem evil and frightening when placed over it.
The horrific ruthlessness of the clone. Surprisingly, according to the monologue in Star Wars Battlefront 2,troopers kind of disliked Ki-Adi-Mundi, but what about Obi-Wan? According to the cartoon, he and Cody were allies and friends for YEARS and he had no hesitation in fulfilling the Order 66. The fact that the clones could be your friends, but be absolute obedient to the point where they will do anything is huge Paranoia Fuel and just plain disturbing.
As explained on the series Nightmare Fuel page, Star Wars: The Clone Wars manages to make Order 66 even more disturbing by giving us more information on how it works, and it turns out the clones are just as much of victims as the Jedi are. The Order is a result of brainwashing chips implanted into them when they're still embryos and are set to activate on a trigger.
Anakin's implied-but-not-actually-shown murder of the children in the Jedi temple during this scene.
His march to the Jedi Temple (With hundreds of Clones in tow) was not much better. While not directly nightmare-inducing, you just tell just by watching (And thanks to Mr. John Williams, hearing) that things are only going to get worse...
If you think the image shown above is scary, than we personally must advise you: don't look up the behind-the-scenes photos of the makeup of Anakin's burn wounds. We're really warning you...
Consider this: for many children, both Revenge of the Sith and The Empire Strikes Back are probably the first films they've ever seen in which the bad guys actually win; the heroes are either dead or have barely escaped with their lives.
Minor in comparison to the ones above, but in the beginning battle when Obi-Wan and Anakin are fighting their way onto Grievous's command ship. They get attacked by a group of Buzz Droids, tiny droids that like to tear things (including ships and droids) apart. R2 is able to fight them off pretty easy, but R4 ends up getting it's head torn off and sent flying into space squealing all the way. Obi-Wan looks quite perturbed after having witnessed it.
Obi-Wan: Oh, dear...
When Anakin kills Dooku, he's highly conflicted about it. When Vader kills the council in the book, he doesn't bat an eye. In fact, he makes little quips about it. Remarkably funny quips.
The death of General Grievous was pretty graphic for a PG-13 film. First, Obi-Wan rips his chest plating open, exposing the working organs suspended in life-preserving gel. After he gets thrown away, Obi-Wan uses his Jedi powers to snatch one of the General's heavy blasters. He fires a shot right into his chest, the General reaching down to grasp at the now burning organ sack, before Obi Wan keeps firing. By the end of his shots, the General has jets of flames shooting out of his eye sockets before he crashes to the deck, every organic part in his body charred to ashes.