Neutral attacks (officially known as standard attacks), attacks performed by pressing the button(s) assigned to the attack function (the A button in most control schemes) while one's character is standing still and the control stick or d-pad is in a neutral position.
Smash attacks, chargeable (cannot be charged in Super Smash Bros.) attacks performed by holding the attack button, then "tapping" the control stick/D-pad as the button is released. In Brawl, the control stick can be pushed in order to execute an immediate smash attack in any of the four regular directions.
Aerial attacks (officially air attacks or midair attacks) any attack performed in midair (using the A button) that is not a special move.
Tilt attacks (officially known as strong attacks), ground attacks performed by lightly pushing and holding, or "tilting", the control stick or D-pad in a direction and pressing the attack button. Attacking while the character is walking also results in a tilt attack.
Special moves, whereof some are special attacks, which are attacks activated by pressing the special move button (the B button in most control schemes) along with a direction. Note that not all special moves are attacks.