The first thing I want to address to everyone with this - keep it clean. Shipping can be included, but only if it isn't overdone. What do I mean? Edited clips where there is a bit of .. suggestive material going on - keep that out of here.
That said keep it clean and link us some cool vids. If you would, give a description of the vid and your opinion of it as you post a link! Don't just leave us links to click and wonder where we're going. :)
To start it off; here is a video I really admire that was designed specifically for the character Katara. "Stand in the Rain". It was not designed as a means of shipping, and actually displays both moments with Zuko and Aang; although it's clearly showing her interest for Aang these clips are very short - and not the point of the video!
The vid with "Stand in the Rain" depicts Katara's difficult life and her determination to never back down, but to stand no matter the cost. My most favorite part is the semi-acapella portion 2/3 through the song... watch!