HAL is generally creepy all throughout the film, but the sequence where he murders Frank Poole by ramming him with a space pod, severing his oxygen tube is outright terrifying, not least because it occurs both offscreen and in complete silence. Not to mention the extremely quick series of jump cuts into the lens of HAL's camera before Dave sees Frank go flying past the screen.
Even moreso, Frank's side of the experience. He's doing a routine repair job, when suddenly he is thrown out into space, his air hose torn. He can hear the flow of air get slower...and slower...as it gets colder in his suit...and harder to breath...
That's actually a different scene, where HAL kills off the astronauts in hibernation. Not that that's much better...
Related to this, some viewers noticed during HAL's game of chess with Poole, that HAL is cheating, announcing a wrong turn and even declaring victory when not in a position to do so. The computer they are trusting their lives on is able to lie! Possibly a foreshadowing that something's not right with HAL.
Alternately, Hal could be testing his ability to lie, as per his orders to not tell Bowman and Poole about the real intent of the mission. His primary function was to provide accurate information completely free from error, and as such had never lied before. He could not ignore his orders, but, given his subsequent actions towards the crew, nor could he stand lying. Distorting information, to Hal, must've been like trying to drown himself. He held his breath for as long as he could, and then desperately searched for a way out. He continued to lie out of necessity, figuring if he couldn't wholly stop it right then, at the very least he could end it sooner.
What really sells this is the way he says "I'm afraid". As he's saying it, you naturally assume (given the way he's been talking so far) that he's saying something condescending like "I'm afraid this won't work", or "I'm afraid you're overreacting, Dave", but instead he catches you off-guard by saying only those two words. A simple and piteous expression of fear. From a machine. The effect is nothing short of spine-chilling, on several levels.
His nearly silent, ruthless murder of the three hibernating crewmembers also definitely counts. "LIFE FUNCTIONS TERMINATED" will haunt your dreams.
The approach to the Monolith is one of the eeriest/creepiest scenes in film.
The famous "Beyond the Infinite" sequence is more awe-inspiring than scary, but when it's intercut with images of Dave FREAKING OUT, it gets really creepy.
It gets even creepier before that. There is a momentary sequence where Dave, wide-eyed in awe (to the point that his eyes are bulging out of his sockets) slowly starts to recoil from the intensity of the environment. The lighting and warping that continuously gets brighter and becomes increasingly distorted transforms Dave into this blue-ish, monstrous, yet somehow still human alien-thing. The sequence starts at around 0:40 and goes to 1:27 : http://www.youtube.com/watch?v=YbLRzabppus
Actually, the scene goes on for quite a lot longer than that, lovingly extended in true Kubrik fashion to exactly the amount of time it takes for you, the viewer, to slowly go insane right alongside Bowman.
Dave very slowly killing HAL by removing his memory. HAL's Creepy Monotone is enhanced when you realize that he is terrified and he can't even express it.
It's made a bit more unnerving when you realize that Dave isn't so much killing HAL as he is lobotomizing him, and the fact that you can't quite tell if HAL is just trying to trick Dave by appealing to his more emotional side or if he's genuinely scared.
The reason why Dave is going to the monolith. On one hand it's obviously because he wants to find out what is up with the thing, that the government was willing to give HAL conflicting orders that led to the death of the rest of the crew. But on the other hand it's because he is essentialy stranded, since without HAL or the rest of the crew he can't get the Discovery back to Earth, he is doomed either way.
That damn Star Child at the end of the movie is pretty creepy, especially when he looks right at the audience. This troper was 13 the first time he saw it and didn't watch the movie again until college.
The bit with the monkey crushing the tapir skeleton with a club. This is the discovery of tools. If the music doesn't make it clear, it's supposed to be a singular leap forward for life on this planet. But then we realize it's just a more efficient way to break stuff and kill other monkeys.