There are a great many memorable scenes in that film, but for starters I picked this one. The movie is set in the middle of the Cold War, and the US maintains aloft permanently scores of B-52 bombers carrying H-bombs, which can reach Soviet territory within two hours if necessary. Dr. Strangelove has several parallel running threads which are all equally important; one of these is about an actual B-52 crew, commanded by a Major Kong (played by Slim Pickens).
In the following clip, the crew receives a coded message to prepare for an all-out nuclear strike on a Soviet missile base.
Dr. Strangelove is actually a dark comedy and has many instances where you can't help but smirk, giggle or burst out with laughter. But this clip is rather serious, and gives some idea of what it must have been like for those crews, spending hours high aloft, waiting for orders that might involve... starting a nuclear holocaust. What's funny though, and what is only getting more hilarious as the movie progresses, is Major Kong's character, with an inimitable Slim Pickens (real name Louis Burton Lindley, Jr.), whose Oklahoman/Texan drawl lends a burlesque quality to the dead serious instructions he has to give his crew.
For those of you who haven't already, discover this gem in its totality.