As boring as it sounds, I think I have to go with just regular old wavedashing.
I think it changed the game the most out of all the advanced techniques I can think of. L-cancelling is great, but it was in 64 and I wouldn't exactly call it "advanced."
Wavedashing is what took Melee and propelled it to the next level and opened up a whole other world of gameplay. The game is so very based on movement and wavedashing is an incredible movement tool, especially when weaved in with other movement tricks. We took an accident and cared for it, nurtured it, until it defined entire characters. And wavedash OoS? Come on! How cool.
Casuals will be blown away if you start doing it, it's visually impressive (on some characters). It gives you an instant leg up against an opponent (if you're good at it -- that is, knowing when to use it). It is unique to Smash, as far as I know. It has become synonymous with competitive play. It's just so damn cool.
Also waveshine -_-