A-sticking, also known as T-sticking, is a controller configuration where the C-Stick (right Control Stick) is set to Attack instead of Smash. A-sticking is derived from the C-Stick being set to a different function, similar to B-sticking, and it is a shortcut for standard attacks, hence the A; T-sticking is derived from the stick being set to Tilts.
These are techniques that are possible with the A-Stick setup.
Central movement (Brawl and SSB4)
If the player tilts the C-Stick in a diagonal direction in midair, the character's neutral aerial will be executed. Since this doesn't require the player to have the left control stick in any specific position, they can move horizontally and perform their neutral aerial at the same time.
Instant dash attack
If the player tilts the C-Stick in any direction right after they start their dash, they can start up a dash attack during the frames of the initial dash animation instead of their character's forward smash. Since doing this performs a dash attack, they can cancel it into an up smash in Brawl as well. With the C-Stick set to Smash, it is necessary to tilt the C-stick downward in order to perform an instant dash attack in Brawl and SSB4, as tilting it forward can result in a forward smash, and tilting it upward can result in an up smash. In Ultimate, the instant dash attack requires the C-Stick to be tilted in the same direction the character is facing, regardless of the player's control scheme.
No fast fall down aerial (Brawl)
In both Super Smash Bros. Melee and Super Smash Bros. Brawl, if the player uses a down aerial by tapping the Control Stick down and pressing the A button, they fast fall as they do it. In Melee, players can instead just use the C-stick to not bother having to lightly tilt the Control Stick, as it allows them to execute the down aerial without fastfalling regardless of whether they tapped the C-stick or not. If the C-Stick is set to Smash in Brawl, though, they will fast fall when doing the down aerial, even if it is tilted lightly. To avoid this then, players can set the C-stick to Attack. In Super Smash Bros. for Wii U, the C-stick no longer causes the player to fastfall when executing a down aerial with it while set to Smash. In Ultimate, the player must lightly tap the C-Stick in order to perform a down aerial without fast falling if the C-Stick is set to Smash.
Attack in midair without losing momentum (Wii U)
In Super Smash Bros. for Wii U, presumably due to a bug involving the C-stick's new ability to charge smash attacks by itself, having it held in any direction while set to Smash will cause the player to lose their momentum (in midair) and be unable to move, attack or grab (inputting a grab will cause the character to shield instead) until the stick is released. This prevents the C-stick from effectively working as a substitute to the Control Stick + A Button to perform aerials, as it will now cause the player to lose momentum, even if the C-stick is held in the direction the player is moving. While the C-stick can be quickly thrusted and returned to its neutral position, it does not provide any significant advantage over the Control Stick + A method, since both will now cause the player to lose momentum.
As such, setting the C-stick to Attack prevents this from happening, making A-sticking the optimal choice for players to efficiently use aerial attacks while keeping their midair momentum, like in the previous games. In Super Smash Bros. for Nintendo 3DS, the C-stick on the New Nintendo 3DS is always set to Smash without a possibility of customization, causing this problem to always occur in said game. This issue was fixed in Ultimate.