That style of gun is called a carbine. Carbines are long-handled guns that are bigger than pistols, but smaller than a full rifle. They were popular in the Old West because they're much easier to shoot from horseback than a full rifle.

Check out the 1855 revolver carbine and the 1873 revolver carbine, two popular revolver rifles of the era.

Citations: Guns on the Early Frontiers: A History of Firearms from Colonial Times to the Western Fur Trade, by Carl P. Russel. Carbines are described starting on page 167.

According to the imfdb, that weapon is a "Buntline special". According to Wikipedia, the authenticity of that particular weapon is dubious, but there did exist similar weapons, like the S&W 320 Revolving Rifle.