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I'm seeing a lot of people being confused about these things, so I'll just go ahead and explain them.
L-Cancelling:
When you do an attack in the air when you land on the ground afterward there is a certain amount of time where you have to wait for you character to go through a landing animation. The best example of this may be Link's down air where he does his iconic down thrust from Zelda II. It's a great move if it hits, but if you miss and hit the ground Link spends a long time pulling his sword out of the ground, wasting your time and leaving you open to an attack. However, if you press the L (or R) button right before you hit the ground the time it takes for Link to pull his sword back out is cut in half. This is true for all characters and all aerial moves and there is never a reason to not do it, it will always help you. Want to attack your opponent again immediately after you hit them with an air attack? L-Cancel the air move. Even fast characters benefit from L-cancelling. There is a more advanced technique called SHFFLing (short hop, fast fall, L cancel), pronounced shuffling, that uses L cancelling to hit your opponent quickly with an air attack close to the ground, which, depending on the move you used, can be followed up by another move. After you learn to L-Cancel, every character will feel significantly faster, even the slow characters will start to feel fast (until you remember the fast characters are still faster).
Wavedashing/wavelanding:
While being the most iconic technique of competitive Melee wavedashing is not the super technique it's made out to be. Don't get me wrong, it's really useful and anyone who wants to play competitively needs to learn it, but wavedashing is more of a situational move, albeit with many useful situations. A wavedash is performed by jumping from the ground and then quickly air dodging diagonally down back into the ground (a waveland is the same thing but when you're already in the air and thus don't need the first jump). This will make your character quickly scoot along the ground. And that's it. That's all it does. But this is a very useful way to move in the right situation. Really the hard part about wavedashing isn't doing it, it's knowing when to do it. I'm still figuring that out myself. There are a lot of uses for it though. You could dash (the normal kind) toward your opponent then wavedash back when you get close to try to bait an attack. Wavelanding is useful to get somewhere a little faster when you land on the ground, and can catch an opponent by surprise since they might not expect you to reach them so quickly. You can also wavedash backward off of the stage to quickly grab the ledge (if you're on it your opponent can't grab it).
If you want to know more about techniques in melee just watch the video series Advaced How to Play SSBM I linked, and if you REALLY want to get into competitive smash bros check out SmashBoards, the biggest competitive smash forum. Hope this helps.