Second Jump Recovery
A Second Jump Recovery (abbreviated SJR) is an advanced technique in Super Smash Bros. Brawl, Super Smash Bros 4, and Super Smash Bros Ultimate which takes advantage of a transitional animation frame to replenish the player's midair jump(s). The technique, which can be performed using any character, is most prominent in Brawl.
In Brawl, there is a single transitional frame between animations. This single animation frame usually has no impact on gameplay, but if the player is in the air and later lands on the ground on this one frame, they will be in a strange ground/aerial state. The game treats the player as they are both in the air and touching the ground, so the game fully restores the player's double jump(s). The player is in this state for one frame only, during which they can either perform a double jump, use a special move, or use an aerial/special with a landing hitbox. The main way this can be set up is by perfectly timing an aerial or an air dodge so that their animation ends a frame before the player lands.
As an SJR allows the player to regain their double jump, it also allows the player to set up another SJR which they can do as many times as they want to. This is known as an Infinite Second Jump Recovery (abbreviated ISJR). Most characters in Brawl are capable of performing an ISJR, although there are a couple of exceptions, such as Ike.
Simultaneous double jump and action
The SJR allows a player to act the first frame of their double jump, rather than having to double jump for a frame first. The SJR thus allows the player to double jump and perform another action at the same time (if both actions are buffered). While this sounds like a rather minor difference and this can be performed without the use of SJR (by simply buffering a double jump and another action at the end of any animation), there are a couple of instances where using SJR yields interesting effects:
The main limitation with this property is that it only works with aerials or air dodges that cannot be interrupted. If an aerial has interruptible frames, the next animation used will override the jump, even if the player times it to occur on the same frame as the jump. An SJR works on an aerial's entire animation and not when the player becomes actionable, so the player can only perform the technique when the prior animation is completely finished.
Pikachu and Bowser
While the universal SJR method is highly difficult to perform, Pikachu and Bowser can easily go into an SJR state from particular special moves, allowing them to abuse SJRs far more than any other character.
If Pikachu performs Quick Attack into the ground, Pikachu will be put into the transition animation frame after the teleport has finished. This allows Pikachu to easily Quick Attack into the ground, double jump (although this is optional) and Quick Attack again, from which Pikachu can repeat the process. This is commonly known as Quick Attack Canceling (QAC), and it is one of the most useful techniques in Pikachu's arsenal. Differently from a traditional SJR, Pikachu can set up a QAC anywhere (as long as the player is close enough to the ground) without having to worry about precise timing or spacing. Combined with the move having a useful hitbox, QAC is far more potent than any traditional SJR.
If Bowser lands at any point during his aerial Flying Slam, he will go into the transition animation frame before landing. This gives Bowser his double jump back and allows him to repeat the technique. This is commonly known as Koopa Hopping, and while not quite as potent as Quick Attack Canceling, it is still far more useful and practical than any traditional SJR for the same reasons.
In Smash 4 and Ultimate
In Super Smash Bros 4, the transition animation frame was altered. The player is now considered to be in a completely grounded state, so if they use a ground attack or dodge, they will perform one of those actions. More importantly though, if the player jumps, they will perform their normal ground jump. This removes the player's ability to perform an SJR and it removes all SJR related techniques that existed in Brawl. This animation frame can still be abused to allow the player to perform a completely lagless landing, but the ability to perform a lagless jump has been removed. The buffer system was also altered to remove the ability to perform aerials/air dodges during a double jump's first frame. This was restored in Ultimate, although it became a default feature for double jumps rather than being a quirk of the buffer system.
Despite this, a few special moves still have ending frames which function like Brawl's transition animation frame, allowing certain characters to perform an SJR. Examples include Corrin with Dragon Lunge and Wolf with his Reflector. In these instances, the technique retains its rather high difficulty from Brawl, and it overall still remains a situational and not highly useful or practical technique.
Quick Attack Canceling also returns in Smash 4, though it can no longer be performed on flat surfaces and is thus less useful overall.
Despite sounding fairly useful, the universal SJR and ISJR are quite limited overall and not highly useful or common in competitive play due to the difficulties of performing the techniques. The trick is essentially frame-perfect, as the player has to finish an animation and then land on an exact frame. The second double jump can be buffered, but the initial aerial or air dodge has to be performed on the frame in order to perform the technique. This is especially hard to do when the player is regularly falling, as the player not only has to perfectly time the aerial, but also has to perform the correct animation while considering factors such as platforms and fast falling. Some characters do have SJR setups using their regular jumps, but even these setups typically require at least one frame-perfect input. ISJR setups also usually require at least one frame-perfect input, with no characters having a bufferable SJR setup, making it incredibly hard to chain together multiple SJRs.
This video showcases various SJR setups in Brawl with Captain Falcon.