A SMAAAASH!!, or a SMAAAASHing hit as described by a Friendly Mole, is effectively the Mother series's term for a critical hit; it is a random occurrence in which an attack deals about twice as much damage. This can only occur when using a character's normal attack, such as "Bash" or "Bite".
In Mother, Smash Attacks were a rare and powerful move that happened at random, completely ignoring the target's Defense. A well-known glitch can occur when getting a Smash Attack, instead of being the normal color, it might accidentally use an enemy's color palette (i.e. Mr. Bat = Purple SMAAAASH!!). This only occurs in the NES version, as in Mother 1 + 2 it is always green.
The chance both to deal and to prevent a SMAAAASH!! attack in Mother is affected by the stat Fight.
In EarthBound Edit
When a character in the player's party deals such an attack, the screen flashes green twice and "SMAAAASH!!" appears in the text box on the upper side of the screen. If a character from the player's party receives a SMAAAASH!! attack, the screen will instead flash red once.
The chance of dealing a SMAAAASH!! attack in EarthBound is affected by the character's Guts. Surprisingly, the Luck stat has nothing to do with it. Jeff cannot SMAAAASH!! in the SNES and Super Famicom version, seeing as his means of attack involve long ranged weapons. However, in Mother 1+2, Jeff can hit a smash. This is probably an error.
SMAAAASH!! attacks in Mother 3 work the same way that they do in EarthBound, except that "SMAAAASH!!" appears above the victim's sprite if dealt to an enemy, and above the character's HP meter if dealt to the player's party.