The Omen/Nightmare Fuel

Jerry Goldsmith's classic Ominous Latin Chanting is terrifying alone. Then they add it to scenes featuring a Creepy Child Anti Christ and people being murdered.
Damien. The concept of the Antichrist being a child who is fully aware of the terror that he is directly and indirectly causing is freaky. What could make that worse? The fact that he acts so innocent and adorable most of the time.
The evil Rottweiliers.
What is it with the nannies in this film? One's hanging herself to death during a party, admittedly it was during the devil's son party, so it was probably the perfect gift. Then we get a new one with eyes so evil no parent in their right mind would allow her near the house never mind their child.

Evil Nanny: Have no fear little one. I am here to protect thee.
Damien: *smiles*

It's creepy enough looking at an ancient painting showing the face of the Antichrist, it would be even worse when he looks just like your adopted son.
The ironing scene in the third movie. The fact that we don't actually see Dean's child getting ironed to death really doesn't help.
Thoughout the first three films, it becomes clear Damien's followers are everywhere and could literally be anyone.
The most terrifying moment is at the end of the first film where Robert Thorne goes into the church dragging Damien inside in order to kill him by stabbing him with knives. All the while Damien is screaming "No, Daddy, no!". Yes, it's the Anti Christ, but still...
Keith Jennings getting his head lopped off. The original version is arguably more frightening due to the film makers pulling it off convincingly without the aid of special effects.
Then there's the graveyard scene, when Robert Thorne digs up the grave where Damien's real mother lies, only to find the skeleton of a jackal resting in there. He then digs up the child's grave, with the logic that if it was the skeleton of a jackal cub then his real son was still alive. But upon finding a human baby skeleton inside, the sorrow on his face is painful to watch.

"They killed him the instant he was born."

The scene in the 2006 version when Kate is in the hospital and the evil nanny uses a syringe to create an air bubble in the IV line which slowly moves down the tube and Kate is unable to do ANYTHING. The original just has the evil nanny chuck Kate out the window, which is kind of anti-climatic compared to the other deaths in the movie.
The scene with the troop of baboons going crazy when they realize who Damien really is.
Damien killing Mark via a brain hemorrhage in the second film when the latter refuses to join Damien's quest for world domination.
Most scenes involving Mrs. Baylock were this for me. Her ability to seem so collected and proper for one scene, than outright demonic in another. The best example is the climax where she sneaks up on Mr. Thorn and starts biting and clawing at him and she just won't go down. It takes two knives in the neck just to stop her. Or a car if you're going by the remake.
From the fourth movie, when this woman was bitten by snakes and died under the shadow of the upside-down cross? The music did not help at all.
Also from the fourth movie, when that detective was hearing people singing Christmas carols as he saw the little Christmas statues of the baby Jesus, this happened: he gets a vision of the baby Jesus statue turning demonic and the carolers turning into creepy Satanists, chanting a spooky chant in Latin and flipping a cross upside down. It might make one decide to never watch The Omen ever again.