A-sticking is a controller configuration where the C-Stick (right control stick) is set to Attack instead of Smash. A-sticking got its name from the fact that the C-Stick is being set to a different function, similar to B-sticking, and that it's a shortcut for standard attacks, hence the A.
These are techniques that are possible with the A-Stick setup.
 Central movement
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 start up dash animation instead of their forward smash. Since doing this performs a dash attack, they can cancel it into an up smash as well.
 No fast fall down aerial
In Super Smash Bros. Melee, if the player used a down aerial with the control stick + A button method, they would fast fall as they did it. This returns in Brawl if the player is using a Gamecube controller. However, in Melee, if the player used the C-Stick for a down aerial, they would not fast fall. If the C-Stick is set to smash in Super Smash Bros. Brawl, then they will fast fall when doing the down aerial. However, if it is set to attack, they will not fast fall.