Well the obvious ones that have already been mentioned first:
"War...war never changes."
-Ron Perlman, Fallout
"So wake up, Mr. Freeman. Wake up and smell the ashes."
-G-man, Half-Life 2
"Hello there! Welcome to the world of Pokemon!"
-Professor Oak, Pokemon (duh)
"But you had something they didn't, something no one saw but me. Can you guess? Luck."
-Cortana, Halo 3
And now a couple that haven't been mentioned:
(Even though I can't quote it word for word, it's still memorable, setting the tone for the rest of the game.)
"Stay a while and listen, and I will tell you a story. A story of Dungeons and Dragons, of Orcs and Goblins, of Ghouls and Ghosts, of Kings and Quests, but most importantly -- of Wizards and Vamp -- Well... A story of Wizards."
-Vlad, Magicka
And not so much a quote, but the opening of Borderlands when Cage the Elephant's "Ain't No Rest for the Wicked" really starts to pick-up, after Marcus's van hits the skag.
If you want a quote, then how about this:
"So... You want to hear a story, eh? One about treasure hunters? Haha, have I got a story for you!"
-Marcus Kincaid, Borderlands