Any presentation of a Lifetime Achievement Award can count, but especially Charlie Chaplin's in 1972.
From 1975: Louise Fletcher giving part of her acceptance speech for One Flew Over the Cuckoo's Nest in American Sign Language for the benefit of her parents, who were both deaf.
From 1990: Daniel Day-Lewis's first Best Actor acceptance speech for My Left Foot, where he made sure to give special attention to Hugh O'Conor, who had played Christy Brown as a young boy and was just as convincing.
Also, the reaction to the rest of his nominees to the announcement of Day-Lewis as the winner. Unusually, they were ALL ecstatic. Kenneth Branagh and Robin Williams in particular look like they're about to leap out of their seats instantly. The crowd ended up giving Daniel a standing ovation, particularly notable for an actor who was hardly that well known.
Also from 1994: In one of the cutest moments in Oscar history, Anna Paquin's reaction to winning Best Supporting Actress for The Piano (skip to 0:54 or 0:55 and look at her face when she realizes that she's won). She was only 11 years old at the time and you can tell she really did NOT expect to win. After being handed her statuette, Anna just stood at the podium hyperventilating and giggling for about 20 seconds, like the little kid she was at the time. Even the audience in their seats couldn't help but laugh at how adorable she is.
Extra points for how genuinely happy Holly Hunter and Emma Thompson were to see Anna win. To clarify, Hunter and Thompson were also nominated in that category, and the former would win Best Actress for the same film later that evening.
From 2002: Halle Berry's tearful acceptance speech after winning Best Actress for Monster's Ball, being the first African-American woman to hold that honor.
From 2005: Jamie Foxx's speech after winning Best Actor for Ray, especially when he choked up speaking of his deceased grandmother, who he said was his first acting teacher. Check it out for yourself.
From 2008: Jon Stewart bringing Markéta Irglová back onstage so she could give her acceptance speech for "Falling Slowly" as Best Song, after she was played off before she could say anything.
That was actually the late Oscar producer Gil Cates's idea to bring her back. He found out that the crew had mistakenly given the cue to the orchestra, so during the commercial break, he asked Stewart to let her come back on stage and finish her speech.
The win by Henry Fonda for Best Actor in On Golden Pond, made all the more notable because he was at home, and his daughter and co-star Jane Fonda (herself a Hollywood star and Oscar-winner) accepted it on his behalf. The experience is made all the more heartwarming when one considers the context of the situation. Henry and Jane hadn't been particularly close for the longest time, and it was while filming the movie that they reconciled. Jane has mentioned that accepting the Oscar on behalf of his father was one of the more emotional of her life, given how it represented the ultimate acknowledgement of her father by his peers.
At the 2014 Academy Awards a week after Harold Ramis' death, Bill Murray presented the award for Best Cinematography with Amy Adams. After saying the nominees and before announcing the winner, Murray went off script and said: "Oh we forgot one. Harold Ramis. For Caddyshack, Ghostbusters and Groundhog Day", and the audience at the ceremony applauded.
Even more so: Murray and Ramis had some very bad arguments while making Groundhog Day, and never so much as spoke to each other again after filming wrapped. But they had reconciled shortly before Ramis' death, and this shows he was willing to put that bad blood aside and properly honor his old friend.
Seeing Heath Ledger's parents and sister accept his posthumous Best Supporting Actor prize for The Dark Knight, and absorbing the crowd's reaction. The ultimate bittersweet moment, in honor of an actor taken from us far too quickly.
At the 1965 Academy Awards, Audrey Hepburn presented Rex Harrison the Best Actor Oscar, for My Fair Lady. In his acceptance speech, he offered to give her half of the Oscar, to compensate for her lack of a Best Actress nomination. He closed by expressing a "deep love" for both Audrey and Julie Andrews (Eliza Doolitle in the original Broadway production of My Fair Lady), his "two fair ladies".