In gaming, to dual wield is to hold a weapon in each hand. Dual wield may be called akimbo. This most commonly refers to matched pairs of handguns (many first person shooters have such weapons as "Berettas akimbo" or "pistols akimbo") but can refer to any other weapon that can be held in one hand such as machine pistols (e.g. "akimbo MAC-10s ") and even melee weapons (e.g. katanas or lightsabers). You don't always have to dual wield two of the same weapons.
Dual wielding greatly decreases your accuracy. You might not be able to look down the sights, knife people or carry a shield. However, dual-wielding typically increases damage inflicted. Also, given that the player-character usually holds both weapons in front of them, dual wielding makes you feel like a total bad-ass.
Dual wielding was largely popularized by Hong Kong action cinema, particularly the films of John Woo and Chow Yun Fat, before making their way into Hollywood movies and eventually action games.
Many early games required the player to pick up a second weapon of the same type from a defeated enemy who had been dual wielding. For example, in Goldeneye 007, the player could only dual wield phantoms after picking the weapon up from an enemy who had been dual-wielding phantoms.
Other games, such as the Tomb Raider-series, feature a protagonist who dual wields by default. When a new weapon that can be dual wielded is acquired, they are picked up in pairs.
Some more modern shooters allow the player to mix and match weapons to maximize damage output.
There are two schools of thought when it comes to dual-wielding in modern shooters. In Left 4 Dead, for example, when the player picks up a second pistol, the effectiveness of the pistol as a weapon class is increased. Two is better than one. Accuracy is not sacrificed. In Left 4 Dead 2, the player could opt for a single Magnum instead of two semi-automatic pistols, sacrificing speed for damage. However two semi-automatics is still beter than one semi-automatic.
The other school of though adopts a cost/benefit system. In the Halo-franchise, dual wielding sacrifices the ability to throw grenades. Many games, such as Knights of the Old Republic, use accuracy or individual weapon damage as a tradeoff.
These two schools of thought often correspond with the two most popular dual wielding control schemes. In Left 4 Dead, a single fire button is used to fire both weapons. With the Halo franchise, the two individual triggers fire the left and right hand weapons respectively.