Equipment is an additional way (aside from levelling or prompting) for you to improve your character's HP and damage in battle! You can get equipment from the Resource generator or from completing certain battles.
You can add equipment to your character by clicking on their icon in the Roster, and then clicking one of the four plus signs next to their picture. Different kinds of equipment will give the character different HP and damage bonuses, but only equipment of the same color can be equipped to a character.