Equipping gear allows you to enhance your Heroes and Captains in fights against monsters or other players. The types of gear available range from utility-based items (e.g., equipment that increases resource production in your City) to battle-oriented items that reinforce specific troop types or your army as a whole. Combining equipment with the Meriones avatar allows you to increase the stats of the equipment by 50% in battles.
Equipment bonuses only apply where they are present. Let's say you send a Hero equipped with gear that increases your guardsmen's attack stats on a march. If you then send out another march led by a Captain to a different battle, the second march won't benefit from the bonus granted by the aforementioned equipment. Here's another example: Equipping your Hero with Crypt exploration equipment won't affect the exploration efficiency of your Captains, because the gear is in the City and not actually being used by your Captains (the only units who can march to explore Crypts).