This is a special battle condition that occurs when a character's HP lands on 7,777 HP.
The character's name will flash in the bottom bar and "All Lucky 7's" will appear at the top.
The character will attack the enemy; hitting it 64 times in succession with each hit dealing 7777 for a total of 497,728 damage. Following the automatic attacks, until the HP changes, attacks made manually will also be 7777 damage.
This can be engineered with clever uses of materia or patiently battling and healing to get the right HP.
This is very useful if you are fighting either of the two strong Weapons; Emerald or Ruby.