Doctor Who: they were showing some episodes on New Years Eve on BBC America, and I caught “The Waters of Mars.” They had me at ‘David Tennant being intense’ and now I’m a hardcore Whovian. 
Along the same lines, altho I saw quite a bit of Three, Four, and Five years back on PBS, the first nu who ep on SyFy, waaaaay back in 2005, was ‘Rose.’ The minute I laid eyes on Christopher Eccleston’s Ninth doctor, all badass in leather, black jeans, and attitude, I was ON BOARD. 
It continued thru the Tennant era, but alas, I’m lukewarm at best over the 11th Doctor at this point. I’ll continue to watch, but it just isn’t quite the same for me.
Battlestar Galactica (the reboot): The opening, first five-minute scene. I’d not heard nor read hardly anything about it, just saw a promo for it and decided, what the heck, and had my TiVo record it. That cold, stark, low-key, nearly dialog-free first scene with Number Six and the bored human bureaucrat, Holy Shit! This was definitely *NOT *the silly, live-action cartoon of my childhood! I was hooked then & there…
The new Battlestar Galactica had a scene in the pilot where the Cylon chick is undercover on the human world, and someone leaves a baby with her for a minute while the go in to a store. She does a little villain soliloquy, and then casually snaps the babies neck and walks away.
Was cool since a) Was a kinda gutsy move for the show. I can’ think of another show where someone explicitly kills an infant on screen. b) It was ambigious, since it wasn’t clear if she killed the kid to spare it suffering the coming Cylon attack or because she hated humans. Made me want to figure out what the deal was.
I gave up on the show after the second season, but it started real strong.
Agreed. Up to that point it appeared to be a typical gritty cop show. Worth watching but not above and beyond the genre as a whole. Then that final scene took it to a whole different level.
The Dancing On My Own scene from Girls. A moment of sincerity in a show about how difficult sincerity can be.
Friday Night Lights, first episode:
The scene where backup QB Matt Saracen has to come into the game after starting QB Jason Street gets paralyzed. The referee calls for captains to meet him on the field. Coach Taylor has to tell Matt that the quarterback is the captain. You can feel the weight of the world quietly settling on Matt’s shoulders at that moment.
The Americans, the episode where Felicity (Elizabeth) beats up Grannie. I’ve never seen such raw hatred coming from a female character on any show.
For me it was actually the second episode (I think) when coach asks Matt if his eyes were open or closed when he made that pass - “My eyes were wide open, coach”. I knew right then I would watch as many episodes as they made.
Breaking Bad: Walt tries to put the broken plate back together
I can’t pin it down to an exact scene, but Buffy grabbed me when I caught a rerun with The Mayor. FX was nice enough to run the episodes in order but I still had some catching up to do.
Once Upon A Time - this one hooked me before I even watched a single episode - it was print ads in a magazine about the new show coming. I am so mad for fantasy and great costumes and magics and fairy tales, I thought, I’m in. And I watched the first couple of episodes perfectly contented, but when I saw Robert Carlyle as the snaggle-toothed half mad Rumplestiltskin and the more civilized Mr. Gold - OMG, I MADE time on Sunday nights to watch.
I’ll third this! It is the only time where I’ve been SO impressed with a show’s first episode that my jaw hit the floor. The best part is that the show remained exceptional.