Two-handed weapons are varieties of hard-hitting weapons like large axes, swords, and hammers. They do less damage compared to two-handed weapons, but they are faster and give you the advantages of having a shield, which include being better able to defend against ranged attacks (such as thrown objects and arrows) and being able to defend yourself better if your horse is killed.
One-handed weapons are relatively light weapons designed for use with a shield. To be an effective hand-to-hand warrior, the player must have a high level of Strength and skills in Power Strike, Shield, Athletics and (optionally) Weapon Master with high weapon proficiencies in any of the Melee Weapons that the proficiency applies to. From horseback, the lance dominates when couched, which delivers the most damage to be found in the game.
Weapons such as the shorter axes are more effective against shields and more capable in close quarters, where as a polearm is (often) near useless in close combat. Axes, swords, polearms, fists, and, in Mount&Blade: Warband, some throwing weapons, can be used in hand-to-hand combat.