This command is two-fold.
When executed, the character "hides" from the battle, never to be targeted again (kinda like how Jump works when Kain is in the air). Statuses like gradual petrification and count are temporarily halted (Jump does the same thing).
Edit: New Info
In the GBA version, the Hide command is automatically executed on the very next turn according to the ATB bar if the character's current HP is less than 1/4 their max HP (which is the definition of being weak). The ATB bar is reset when the Appear command is executed.
In the SNES version, the Hide command is automatically executed when the character is weak, regardless of the current allied turn being executed. This effectively takes the character out of the battle easily unless you manage to heal said character in a really small gap after being told to show themselves. The ATB bar is reset when told to Show, but that doesn't prevent Hide from being executed.
However, this behavior appears to be averted in a boss themed battle (need to complete the battle at Fabul to be sure, perhaps it is the boss bit that is being adhered to instead) such as the Antlion battle.
It is unclear if the GBA version follows the SNES version's behavior regarding the Antlion battle (and other battles). I am sure that the boss bit doesn't change the behavior of Hide though.
We have EVEN MORE themes now, but some old ones are gone...