Description taken from Arcade version of this game

Golden Axe is a variant on the traditional beat-'em-up with a swords-and-sorcery theme instead of urban brawling. Three heroes vow revenge on evil warlord Death Adder for killing their loved ones and abducting the king and his daughter. Each character possesses distinct weapons, special moves, and magic abilities. Choose from barbarian Ax Battler, Amazon warrior Tyris Flare, or dwarf Gilius Thunderhead. Engage waves of skeletons, knights, giants, and other creatures while advancing through five side-scrolling locales. Some enemies ride one of three dragon types that can be mounted by your hero and used as an extra form of attack. Collect and save potions to unleash progressively powerful tiers of fire, lightning, or earth magic, depending on your character. Battle the computer alone or team-up with a friend for cooperative play. ~ Scott Alan Marriott, All Game Guide