Stagger is a state where your character (or the enemy) is reeling backwards and the character can do nothing in that time. Stagger is different from a Poise break, characters in staggered state take extra damage and stagger has longer duration than a Poise break.
A staggered character takes 40% extra damage from everything, including spells (Pyromancies, Sorceries, and Miracles). Hitting a staggered character produces a green hit flash instead of the usual white.
There are many ways a character can become staggered: