Power Shield: If you haven't already noticed, the L and R buttons on the GCN controller are analog. Normally when you need to shield, your first instinct is probably to jam on the button as fast as you can. However, with a bit of practice, you can make use of Power Shielding in situations where you'd normally block regularly.
Simply press the L or R button lightly. If you press it enough, but not too much, you'll notice your shield will be enormous in size and will disintegrate itself rather slowly instead of quickly. This is very useful as you can shield for longer periods of time and be completely covered by the shield longer, but the disadvantage is that it gets destroyed more quickly by attacks.
Note: Be careful when you rest your fingers on the L and R buttons. You may inadvertently start pressing either of those buttons lightly, causing you to shield at inopportune times.

Meteor Smash: A Meteor Smash, simply put, is any attack that sends the opponent downward as opposed to upward or to the side. For the majority of the characters, a Meteor Smash is an aerial down+A attack, but not always. Some characters' Meteor Smashes aren't even aerial attacks.
The most dreaded use of a Meteor Smash is to approach an opponent who's over a pitfall and smash them downwards to their death. If the opponent has enough damage, they won't even have a chance to recover!

Quick Recovery: When you're getting smacked around by a ruthless opponent, often you won't even have a chance to get back up after falling down before getting pummeled some more. Making use of a simple trick, however, you can get up the instant you fall down. After you're attacked and sent flying, if you hit the L or R button as you hit the ground, your character will be cat-like - they'll always land on their feet.

Grapple Recovery: Samus, Link, and Young Link all have grab (Z) attacks that can help them recover in a stylish, albeit tricky, way. When you're jumping back onto the arena as any of the aforementioned fighters, hit the Z button as you're facing the ledge, causing them to shoot their grapple hook/beam into it. They'll fall for a bit, but the Hookshot/Grapple Beam will hold them up. Hit Z to reel yourself up after.

Wall Jump: Captain Falcon, Samus, and Young Link are so nimble that they literally bounce off the walls! With any of those fighters, jump up to a wall, and as soon as you get near it, tap the Control Stick in the opposite direction. Your character will bounce off the wall ala Super Mario 64. If you fall into a tight pitfall, such as those found in Fourside, this is a very effective recovery move as it can be done an unlimited amount of times.

Waveslide: A very odd move has been discovered in SSBM and has been nicknamed the "Waveslide". With any character, hit the X or Y button to jump, but before your feet leave the ground, quickly do a midair dodge, angled downward (Shield and tilt the Control Stick down and to the side). Your fighter will sort of slide along the ground for a bit. Yes, there are practical uses for this, such as moving across the UFO on Fourside and psyching out an opponent who's about to rush you.