Monarchy refers to a system of government in which executive office is held by an individual for life and then passed to their child (or some other blood relative if s/he is childless) upon their death. In some instances, monarchs hold absolute power (see Tsarist Russia); in other cases, they are ceremonial figureheads (as in modern day Britain and Japan). Various states exist in between these extremes.
Monarchy was among the first systems of government to develop and has been around for most of the history of civilization. The reasons for this are simple: someone who got to the top of an early society decided to pass the job on to his son, who later followed this example. Since most trades were hereditary in the past, this was not seen as out of place and became traditional. For the last 200 years, monarchism has been on the decline, being replaced by republics and non-monarchical dictatorships.
Two related concepts are Nobility and Feudalism. Nobles are a class of sub-monarchs: hereditary rulers who oversee smaller areas of territory and pledge their services to their monarch. Feudalism is a system of government based on a monarch and nobles, and it was fairly common historically in various forms. A noble pledges his loyalty to the monarch, or to a higher-ranking noble in one of several tiers of nobility ultimately linked to the reigning monarch. Many monarchical states were forged by one local hereditary leader gaining the allegiance of various local lines in an area. However, it is not universal for monarchical states to be feudal in nature. For example, Imperial China, after the Sui Dynasty, had a centralized monarchy run by bureaucrats selected by examinations, while most Islamic kingdoms ruled through generally non-hereditary military and religious figures, and in the case of the Ottoman Empire, slaves owned by the Sultan.
Monarchies in Fiction
Most fantasy universes involve some form of monarchy, it being the dominant form of government in pre-modern times. While less common, monarchy is not an unusual form of government in science fiction.