How does dual wielding work in Diablo 2?

Barbarians (and Assassins, in the expansion) are the only classes who can dual wield.

Barbarians can dual wield any one-handed melee weapons - one-handed swords, one-handed axes, one-handed hammers. They can also dual wield throwing weapons like javelins, but without the Double Throw skill they can't throw from both hands. As a final point, they can also wield a two-handed sword in a single hand in order to dual-wield. This does reduce the damage, and the power of the weapon for both single-handed and two-handed version is listed on hovering. Assassins can only dual wield their claw weapons.

When you perform normal attacks, you will alternate between the weapons in your hands. The attack speed is calculated for each arm swing, so different weapon speeds will result in different attack speeds. However, the speed is generally faster than if you just swung either of the weapons twice. Clicking faster has no effect, and the damage is simply that of each weapon hitting. And of course, dual wielding is necessary to use a number of combat skills like Double Swing, Double Throw, and Frenzy for the Barbarian, and some martial arts skills for the Assassin.

Your right hand strikes first, but in most scenarios it's fairly moot whether you keep your weapons on one hand or the other.

This calculator will help you determine what kind of effect on attack speed and dps you will get by dual-wielding.