Season 3 mostly due to Jaime's character development. The bathtub scene with him and Brienne in Kissed by Fire is probably my favourite scene in the entire show, and along with being humbled by the loss of his hand and rescuing Brienne from Locke was what changed him from in my eyes being an awful person to my favourite character. Along with this, I thought Jon's story with Ygritte and the Wildlings was a really interesting change from the previous two seasons, and seeing another side to the Wildlings was awesome, and the conflict inside Jon to do his duty or be free is great. Plus Jon+Ygritte c:
Also, Daenerys starting to actually do stuff is, again, a nice change from S1/2 where she kinda just sat back and went with the flow of things instead of being a real Queen. The ending of And Now His Watch Is Ended is badass, and reintroducing Barristan and bringing in new characters like Grey Worm and Daario is really exciting. The King's Landing stuff is very interesting too, seeing Tywin be more and more awesome, Tyrion deal with no longer being Hand, and Margaery subtly controlling Joffrey (and Cersei realising this).
There isn't much to mention with Bran's story, mainly because whilst it's kinda interesting, not much ever happens. Introducing Jojen and Meera is cool though, and Bran starting to realise his powers and his destiny. Robb's story is heartbreaking this season though, as obviously the Red Wedding happens in The Rains of Castamere and as much as I love him he makes some awful decisions. Theon's story is also horrible to watch, but similar to Jaime's redeemed him in my eyes from his mistakes in Season 2 (although I don't think I can forgive him for Rodrik), with "My real father lost his head in King's Landing".
Lastly Arya's story in S3 is really cool, introducing Beric Dondarrion (who is just awesome) and getting some more character development with the Hound as they travel together, losing Gendry to Melisandre and of course seeing the results of the Red Wedding.
Sorry for the giant wall of text but Season 3 is definitely my favourite season so far, it was crazy fantastic and just brilliant. Special shoutout to Season 2, who after rewatching recently is definitely my second favourite season, with episodes like Blackwater and Garden of Bones. As for favourite episodes, I'd have to say Kissed by Fire, as well as Blackwater and Watchers on the Wall.