A Special Move is a move performed when pressing the B button, and is a part of a moveset. If you do not touch the control stick while pressing the B Button, you will perform your neutral special move. There are three more special moves depending on whether you tilt the control stick to the side, upwards, or downwards. These moves are called Side Special Moves, Up Special Moves, and Down Special Moves. Sometimes instead of the term Special "B" is simply used due to the button pressed to perform the move, such as "Down Special" Or "B Down".
Special Moves are the bread and butter of a moveset, usually being the most unique and interesting moves of a character's moveset. The movesets of Sakurai rely heavily on special moves, the rest of the moves being horribly generic. In a more original moveset, the other various attacks, the Smash Attacks in particular, can bear more resemblance to special moves.
An example of a special move:
Wario's Down Special Move: Wario Waft
If one uses this button input at the start of the match, Wario would just waste time letting out a cute little poot. However, as time passes Wario gathers up more gas in his rear end. Shortly before 2 minutes pass, Wario's fart will be fully charged and will do massive knockback and nearly 50% damage. Wario gets propelled up from farting, and he's the hitbox as he skyrockets upwards, the area around him also being a hitbox as he first lets out the fart. The fart actually does more knockback when not fully charged. This is an excellent move for keeping, punishing, KOing, and edge guarding, just make sure you don't waste the charge. In a dire situation, the fart can even be used to recover. The fart has many purposes, but is rarely able to be used due to it's immense charge time.
The Wario Waft fully charged.
Wario let out a cute little poot due to not having held it in.