"What makes for a memorable and awesome experience when playing an RPG?"
The main thing is a good, character-driven story. Games like Mass Effect 2 and Final Fantasy X are great examples of these for having a focus on their characters and inter-relational drama. Other games like Final Fantasy XII fail at this, sometimes delivering an interesting grand narrative but offering little to no character development. An RPG should have an interesting story with a fascinating, detailed fictional world, multi-faceted characters with many layers and secrets, and 40+ hours in which to carefully unfold a twisting, emotional narrative.
"What are some of your favorite RPGs and some reasons why you like them?"
Final Fantasy VI, VII, X, and Mass Effect 2 are easily my favorite RPGs. And I like them for the reasons stated above, pretty much. All have great gameplay as well, and the Final Fantasy soundtracks are also some of the most beautiful works of music ever composed, but the main thing is the story.
I would love to see more RPGs done in similar fashion to the old ones. Even go back to random battles if you want, just take things back to their roots. I'm actually working on a game with a team right now toward this very goal. I hope it works out well.