Weapons are used by your heroes to attack opponents. Each weapon has special properties which makes it good for specific situations: the sword for instance is specialised for attacking single tough opponents, while the bow is great for taking out weaker opponents in the back.
Each hero wears one weapon during a quest, selected using the Weapons menu. Each hero can wear any of the available weapon types (not all weapon types are available at all times, some need to be unlocked) and there is no limitation between heroes; sending out all three heroes with swords is perfectly ok.
It is possible to upgrade a weapon for a hero, resulting in extra bonuses when using the weapon type.