A repeating firearm or repeater is any firearm (either a handgun or long gun) that is capable of being fired repeatedly before having to be manually reloaded with new ammunition from the magazine.
Unlike single-shot firearms, which can only hold and fire a single round of ammunition, a repeating firearm can store multiple cartridges inside a magazine (as in pistols, rifles, or shotguns), a cylinder (as in revolvers), or a belt (as in machine guns), and uses a moving action to manipulate each cartridge into and out of the battery position (within the chamber and in alignment with the bore). This allows the weapon to be discharged in relatively quick succession, repeatedly, before manually reloading the ammunition is needed.
Typically the term "repeaters" refers to the more ubiquitous single-barreled variants. Multiple-barrel firearms such as derringers, pepperbox guns, double-barreled rifles, double-barreled shotguns, combination guns, and volley guns can also hold and fire more than one cartridge (one in each chamber of every barrel) before needing to be reloaded, but do not use magazines for ammunition storage and also lack any moving actions to facilitate ammunition-feeding, which makes them technically just bundled assemblies of multiple single-shot barrels fired in succession and/or simultaneously, therefore they are not considered true repeating firearms despite their functional resemblance. On the contrary, rotary-barrel firearms (e.g. Gatling guns), though also multi-barreled, do use belts and/or magazines with moving actions for feeding ammunition, which allow each barrel to fire repeatedly just like any single-barreled repeater, and therefore still qualify as a type of repeating firearm from a technical view point.
Although repeating flintlock breechloading firearms (e.g. the Lorenzóni repeater, Cookson repeater, and Kalthoff repeater) had been invented as early as the 17th century, the first repeating firearms that received widespread use were revolvers and lever-action repeating rifles in the latter half of the 19th century. These were a significant improvement over the preceding single-shot breechloading guns, as they allowed a much greater rate of fire, as well as a longer interval between reloads for more sustained firing, and the widespread use of metallic cartridges also made reloading these weapons quicker and more convenient. Revolvers became very popular sidearms since its introduction by the Colt's Patent Firearms Manufacturing Company in the mid-1830s, and repeating rifles saw use in the early 1860s during the American Civil War. Repeating pistols were first invented during the 1880s, and became widely adopted in the early 20th century, with important design contributions from inventors such as John Browning and Georg Luger.
The first repeating gun to see military service was actually not a firearm, but an airgun. The Girardoni air rifle, designed by Italian inventor Bartolomeo Girardoni circa 1779 and more famously associated with the Lewis and Clark Expedition into the western region of North America during the early 19th century, it was one of the first guns to make use of a tubular magazine.