Any kind of research into the Old West will inevitably lead you to the "six-shooter" (more properly termed revolver). It's pretty tough to avoid, in fact. It's a staple of this era.
The huge technological advancement that made the revolver popular was its ability to fire multiple rounds without reloading. Just prior to the revolver (this means for the entire American Civil War) guns, including handhelds, could only fire a single round. This means that the wielder had to fire, go through a lengthy reload process, and then fire again.
You can imagine how effective this was in battle (read: not very). It's also why guns were frequently paired with some type of sword or bayonet. If the person fighting couldn't reload fast enough they were going to have to have another weapon option available, fast.
While the revolver as a technology existed before the Civil War, it wasn't until Samuel Colt's percussion cap revolver and, later, Smith & Wesson's cartridge revolver that this style of gun started to have mass-market appeal. The cartridge contains both the bullet and the gunpowder which saves the user a considerable amount of reload time.
In a nutshell: revolvers made the handheld gun an efficient weapon. A person with a revolver no longer needed a gun plus weapons x,y and z to compensate. He just needed a gun. To top it off, a gun was capable of killing just about anything it was pointed out, a useful feature for those that were trying to settle the great unknown with untold number of dangers.
Being small, lightweight, easy to use and deadly, it's no wonder the six-shooter became a trademark of this era.