I will start. My favorite quotes all come from Neloth. They are:
“By Malacath’s toenails, where did that come from?”,
“By Malacath’s hairy knuckles, you’ll pay for that!”
“Dagon’s eyeballs, but you’re a ugly one, aren’t you?”
“Do you need all your toes? I could really use a fresh toe for...nevermind. You aren't moldy enough.”
“Well, my pretties. What shall we do today? Maybe see if you can regenerate your fingers?"
I have a few, so forgive me if they take up much space.
"Sheogorath, Daedric Prince of Madness, at your service."-Sheogorath, introducing himself.
"Sheogorath's madness take you!"-Dunmer, when fighting.
"Filthy n'wah!"-Dumner, when fighting. (possibly not used against other Dunmer)
"Where is that lazy steward of mine? Varona! No, wait, she's dead. Drovas!"-Neloth, after "Reluctant Steward"
"She's dead? How annoying."-Neloth, during "Reluctant Steward"
"More of these things?"-Neloth, encountering a Dwemer Automaton.
"The man makes a dreadful cup of Canis Root Tea."-Neloth, commenting on Drovas, his new steward.
"Do you get to the Cloud District very often? Oh, what am I saying, of course you don't."-Nazeem, commenting on the frequency of the Dragonborn's visits to the Cloud District.
"I fight for the men I've held in my arms, dying on foreign soil! I fight for their wives and children, whose names I heard whispered in their last breath. I fight for we few who did come home, only to find our country full of strangers wearing familiar faces. I fight for my people impoverished to pay the debts of an Empire too weak to rule them, yet brands them criminals for wanting to rule themselves! I fight so that all the fighting I've already done hasn't been for nothing! I fight... because I must."-Ulfric Stormcloak, when first entering the Palace of the Kings, Windhelm.
"Do you not see? Elven Supremacy is the only way!"-Literally every Empire/Thalmor supporter in Skyrim ever.
"What is better, to be born good, or to overcome your evil nature through great effort?"-Paarthurnax, when asked why they shouldn't kill him."
"And so the First Dragonborn, meets the Last Dragonborn, at the summit of Apocrytha."-Miraak, just before the end of "At the Summit of Apocrytha.