Shellshifting is a term that refers to Squirtle's unique dash pivot, and by proxy its advanced uses, in Super Smash Bros Brawl. It gives Squirtle's ground movement significantly enhanced possibilities, much like a wavedash in Melee. There are many applications to shellshifting that greatly improve both Squirtle's offensive and defensive capabilities.
How to shellshift
Shellshifting is simply a reference to Squirtle's dash pivot animation. When Squirtle turns in the middle of a dash, it will retreat into its shell, slide a very small distance in his initial direction, and then very quickly slide back in the new direction over a noticeable distance before exiting its shell and resuming its normal dash. The physics behind the slide at all stages while Squirtle is in its shell are what make shellshifting very useful in combat. Alternatively, players can cancel the return slide by smashing the control stick back to the original, pre-pivot direction just as Squirtle begins to turn, which will stall Squirtle in its shell without him sliding much at all (though it will still be facing in the new direction when it exits its shell).
Standard attacks and aerials
Squirtle can easily perform all its standing and jumping attacks out of a shellshift, increasing their speed and range (primarily in the eyes of the opponent, who may have previously been chasing the fleeing Squirtle). The jab, forward-tilt, and up-tilt attacks in particular are useful at interrupting the opponent should they be approaching, while short-hopped aerials like forward air and neutral airs provide explosive power options for Squirtle, allowing it to once again face its foe in the air as opposed to fighting on the ground.
Squirtle can also perform its smash attacks out of a regular shellshift, but all three of them are much better suited for use with slightly different techniques (see pump slide below, dash pivot cancel).
Quickshifting is the act of cancelling a shellshift in order to gain aerial speed. A quickshift is performed by lightly tilting the analog stick back while dashing and quickly jumping, which gives Squirtle a massive speed boost and is good for approaching with a bair or nair attack. If a quickshift is performed to perfection, the shifting sound should not even be heard, and Squirtle will get an even larger boost forward. This is an extremely effective technique against opponents who try to anticipate Squirtle's quick aerials with shielding, and also very effective for edgeguarding opponents who attempt to airdodge, as Squirtle's speed catches them off-guard when using the quickshift technique. Quickshifting can also be used to quickly short-hop over an opponent while landing close enough to execute a tilt-combo, doing as much as 50% or more damage with its u-tilt (works best against fast-fallers and heavyweights). Also, though it is extremely difficult, quickshifting can be used to do Squirtle's Hydroplane, which lessens the chance of being attacked out of a shellshift while initiating a Hydroplane, and is much more unpredictable, as one of the most noticeable times a Squirtle uses shellshift erratically is when getting ready to perform a Hydroplane. Keep in mind that in order to do a quickshift --> Hydroplane, it is 100% necessary to move from tilting back to tilting up very quickly with the analog stick. The player shouldn't return the analog stick to neutral position, but rather slide it from the back position to up position, which should make quickshift --> Hydroplane much, much easier. Also note that when done correctly, a quickshifted Hydroplane does go slightly further than a regular Hydroplane. The act of quickshifting somewhat resembles jumping as if one were closer to the actual origin to the jump.
Shellshift dancing is simply the act of perfectly chaining Squirtle's dash pivots together, performing each one as soon as the prior one has finished its animation. This gives Squirtle the appearance of sliding back and forth in his shell while barely exiting. This technique is useful at long-range for confusing one's foe, potentially inducing them to give away or whiff their next move, giving Squirtle the opportunity to capitalize on a possible weakness in the approach. Unfortunately, it is inferior to a proper dash dance, which has a much lower range of vulnerability and can be more easily canceled into an attack (the downside of true dash dancing is that it is far more difficult to perform). Another drawback is Brawl's tripping mechanic, which factors in with each pivot and increases Squirtle's risk of tripping mid-match, making even a regular dash dance less useful than it was in Melee.
Shellshift to grab
Squirtle's grab range is average on its own, but combined with a good shellshift, he can surprise opponents by returning with a high-speed, mobile standing grab, increasing the grab's range and decreasing its predictability. It is superior to its running grab due to decreased lag and increased distance, and upon grabbing Squirtle often slides a small distance while holding its opponent, making a down-throw KO at high percents even more likely should Squirtle slide its victim to the end of the ledge.
The Hydroplane is the greatest application of the shellshift technique and is somewhat difficult to pull off perfectly. It allows Squirtle to slide across a tremendous distance (almost all of Final Destination) while charging its smashes. It can also be used to slide other moves such as his grab and tilts. This term usually refers to the Pump Slide.
To initiate a shellshifted pump slide, first do a complete shell shift (the longest possible slide the player can get) and then perform a jump-canceled up smash. If done incorrectly, Squirtle will just slide a small distance and do its up smash (a regular dashing up-smash, or "hyphen smash"). It is easiest to do when Tap Jump is turned "on," as the game will register a canceled jump automatically upon performing the up-smash command. However, it can also be done by pressing the jump button at almost the exact same moment as A (the C-stick should not be used). Although the jump technically registers before the up smash, it should feel as though the player is pressing the buttons at the same time.
If performed correctly, there are a few ways the maneuver can be put to use. One way is to leave the smash uncharged, which will release the move at roughly the middle of Final Destination while Squirtle itself slides the rest of the distance, making for a very quick mid-to-long range KO move. Another way is to charge the smash so that Squirtle takes it with him to the end of the stage, releasing a more powerful eruption of water over a longer distance. Either of these methods can be used to simply give Squirtle some defense as it attempts to cross the stage more quickly than either its dash or its Withdraw move would allow.