Hannibal Lecter's startlingly brutal and utterly brilliant escape at the end.
Clarice shooting Buffalo Bill while he was stalking her in the dark.
The first meeting between Lecter and Clarice. The polite banter. Clarice's unflinching answer about the embarrassing thing Miggs hissed at her. Lecter's professor-like tut-tut when Clarice forces the discussion onto that profiler questionnaire. Lecter's rude attempt to upset the rookie agent, and Clarice's hard retort. And then Lecter tells us about that Census taker who tried to test him: "I ate his liver, with some fava beans and a nice chianti." In those ten minutes, Silence of the Lambs owned that Best Picture Oscar.
The final meeting between Lecter and Clarice as Clarice begs for Buffalo Bill's real name. Lecter insists on the original bargain with her, that she tells him about her past...
And she does, revealing her traumatic experience trying to save lambs from the spring slaughter:
Lecter: You still wake up sometimes, don't you? You wake up in the dark and hear the screaming of the lambs. Clarice: Yes. Lecter: And you think if you save poor Catherine, you could make them stop, don't you? You think if Catherine lives, you won't wake up in the dark ever again to that awful screaming of the lambs... Clarice:(in tears) I don't know. I don't know. Lecter:(closes his eyes: he knows Clarice has given up the deepest secret of herself) Thank you, Clarice. Thank you.
And then Chilton interrupts, before Lecter can tell Clarice the name.
It's made clear that Lecter won't tell Clarice Buffalo Bill's true name, no matter what she tells him. Why? Because he knows that if she finds him on her own, it'll give her the advancement she wants. By not telling her, Lecter ensures that she solves the case herself, effectively cementing her status in the FBI.
Catherine manages to lure Buffalo Bill's poodle down into her prison, thus ensuring he can't kill her for fear she'd hurt the dog. In particular, the way she stands up to him, despite having been thrown in a hole, starved and tortured.
And it won all of those despite being released in February, while most other Oscar successes are usually released in November/December (much closer to the ceremony), and despite being (arguably) a horror movie, a genre that is usually ignored by the academy.
Not a specific moment, but the narration in the book explicitly states that the only reason Lecter was able to fashion his handcuff key (from a paperclip and broken piece of pen, both carelessly left in his cell on two separate occasions) was because both incidents took place on Barney's days off. The man's damn good at what he does.
It's somewhat referenced in the film, as Chilton has ordered Barney out of the cell by the time he's left the pen behind.
It's not in the film, only in the novel: Jack Crawford, while he's a good agent, can be extremely manipulative when he wants something, particularly when using photographs of the crime scenes to get a reluctant person to help. The doctor at Johns Hopkins won't have any of it, and tells Crawford it's a "childish, bullying thing to do."
I was a battle surgeon, Mr. Crawford. Put your picture back in your pocket.
Jonathan Demme has one of the finest directing moments of his career in the buildup to the climax. The scene keeps cutting between Buffalo Bill in his basement and the FBI setting up outside, they ring the bell, he heads to answer it...and Clarice is at the door, having figured out his real location while the other agents got duped.
Anthony Hopkins' performance as Lecter has gone down as one of the greatest villains in film history, and it still remains chilling to this day.
A whole 26 years after the film, Jodie Foster finally played Clarice again in a sketch for The Late Show with Stephen Colbert, and absolutely nails every single nuance of her original performance as if she'd just done it yesterday.