One-Handed is a characteristic of weapon types in all three Diablo games. It is often abbreviated as 1H.
A character may only equip weapons and off-hand items into two hand slots, with one-handed weapons occupying one (main) hand only, and two-handed weapons occupying both slots. With one-handed weapon in the main hand (left slot), a character may equip the other (right slot) hand with a shield, an off-hand only item (such as Source or Quiver) or (for some classes) another one-handed weapon. Having two one-handed weapons equipped is generally called Dual-Wielding.
There are some exceptions to this rule (see Two-handed for the list of those), but generally, a character must choose between dual-wielding, picking one-handed weapon + offhand item, or equipping a single two-handed weapon. Not all one-handed weapons may be taken as off-hand choices.
Classes that can dual-wield:
Crusaders may equip a two-handed weapon in main hand, with a damage penalty. Amazons and Demon Hunters may equip their two-handed ranged weapons together with Arrows, Bolts or Quivers. Diablo II Barbarians are the only class that can equip two two-handed weapons, one in each hand, but both will have a penalty to damage, in fact, using their 'one-handed' profile to calculate damage.
Both one-handed weapons in each hand have their own attack speed, damage and special attributes, but general bonuses (such as core stats) from both weapons stack.
A general rule is that one-handed weapons deal less damage, and can roll less total stat bonuses than their two-handed counterparts, but have a faster attack speed. Some class skills may require a specific weapon type to be equipped, so there is no such thing as a 'best choice' here. It is generally believed, and usually proven true, that having a combination of a one-handed + off-hand item or dual-wielding is more flexible and allows better customization of the game style. Most of the time, bonuses from two weapons or one weapon and a side item are greater than those of a two-handed weapon, leaving the latter's potential use being for attack range or roleplaying purposes, except under extreme circumstances.
Wands, Daggers, Hand Crossbows, Javelins, Scepters, Ceremonial Knives, Fist Weapons, Throwing Weapons, Spears (Diablo III only), Sorceress Orbs and Katars are always one-handed and do not exist in two-handed variants.
Note that Legendary and Set weapons in Diablo III are Unique Equipped, that is, one cannot wield two identical weapons at the same time. While players may stack as many of them in their stash as they want, only one of each items can be actually equipped. This includes the set weapons such as Shenlong's Fist of Legend / Shenlong's Relentless Assault and even Kanai's Cube extracted legendary affixes.