Pilot Episode
"Winter is coming!" - Stark household's mantra
"Can a man still be brave if he's afraid?
That is the only time a man can be brave." - Bran asking his father Eddard (This is from the A Game of Thrones book)
"The man who passes the sentence should swing the sword. If you would take a man's life, you owe it to him to look into his eyes and hear his final words." (This is a quote from the book. This dialogue is in the scene where Eddard kills the deserter from the Wall. I'm not sure if these are the exact words used in the show series.)
"Killing things clears my head." -King Robert Baratheon
"Never forget what you are. The rest of the world will not. Wear it like armor, and it can never be used to hurt you." - Tyrion, speaking to Jon Snow in pilot episode
This is the quoted version from the book:
"Never forget what you are, for surely the world will not. Make it your strength. Then it can never be your weakness. Armor yourself in it, and it will never be used to hurt you."
"All dwarves are bastards in their fathers' eyes". - Tyrion, speaking to Jon Snow in pilot episode
"The things I do for love." - Jaime Lannister, speaking to Queen Cersei