What's your Star Trek 'ship: You have to ask? Kirk/Spock.
Why do you 'ship them: Have you seen them? At all? Their closeness is obvious to anyone who looks; hell, they have a word that describes their relationship...that was made up to describe their relationship. It doesn't get more epic then that.
How you discovered them: I've been watching Star Trek (not just TOS, either, but TNG, DS9, Voyager, Enterprise) since I was four and I always thought Kirk and Spock were best friends, when I was twelve I was convinced they were married and when I was fifteen I discovered slash fic and my perverted brain went 'Heeeeyyyy, remember Star Trek?' and the rest is history, lol. It's only recently that I started writing it, though.
Your favorite fanfic: Ugh, I hate picking and choosing fics. I've got tons saved if anyone wants them. One of things I love about ST fandom is that no matter what I'm in the mood for, I can usually -about 98% of the time- find something that fits.
Favorite moment: Oh, sooooo many. All of The Search for Spock. The huge grin Spock gets on his face at the end of Amok Time when he realizes Jim's still alive. The new movie, the first time they exchange glances for the first time.
Is there a canon quote that sums up your 'ship: Fuck, yeah, there is -tons of them, in fact- but the best, imo, is one of the deleted scenes from '09 (See? Even JJ knows what's what, lol): “You once said being a starship captain was my first, best destiny... If that's true, then yours is to be by my side. If there's any true logic to the universe, we'll end up on that bridge again someday.” -Kirk Prime, Star Trek (2009)
If you want a TOS one, then: “And where do you estimate we belong, Miss Keeler?” -Spock “You? At his side. As if you've always been there, and always will.” -Edith Keeler, City on the Edge of Forever
In closing: That is what K/S is -even more then any other 'ship in any other fandom- destiny, soul mates, best friends, home, everything to each other; t'hy'la. It's lasted fucken decades; it, in fact, is the very first bromance on TV and started slash and fanfic. It's inescapable -even in a 'verse where a mad Romulan time travels, changes everything and destroys planets, Kirk and Spock still find each other. Sure, they might each have their side pieces once in a while, but when it comes down to it, when it matters the most, they always choose each other. In Search For Spock, Kirk risked everything -his life, his crew, his career, the motherfucken Enterprise- because there was a chance that Spock might still be alive; in '09 it's Kirk -not Uhura- who forces Spock to feel things, from the first time they lock gazes at the Kobayashi Maru hearing to the last scene, it's Kirk -not Uhura- who Spock comes back to Starfleet for. That's fact, and that's canon. Argue with that.