If the proposed form of government only holds elections upon the death or resignation of the ruler, it is an absolute monarchy or despotism, with an elected monarch.
If the proposed form of government has regularly scheduled elections (such as every year, every two years, every four years, or every six years), it is a republic.
Some countries that have had similar systems: From about 1290 to about 1618, the Holy Roman Empire was an elective monarchy with 7 electors. During Renaissance and early modern times, Poland was an elective monarchy (elected by a large class of nobles). Neither the HRE nor Poland had anything like an absolute monarchy, though. Haiti under the Duvaliers had an elected president for life.
In countries with elective monarchies, a major historical theme is that the ruler seeks to choose his (or her) successor. Often the ruler manages to change the law of succession, either to replace the elective monarchy with a hereditary monarchy, or to restrict the list of eligible candidates to members of his (or her) family. Alternatively, the ruler can designate their successor while they are still in power. This is the purpose of "crown princes", and of Mexico's tradition of the President choosing his party's candidate at the next presidential election.
This form of government can be used to legitimize the results of succession wars that occur after the previous monarch dies. The winner of the war becomes the "winner of the election" instead of a mere usurper. During early modern times, the Ottoman Empire and the Mughal Empire had a tradition of succession wars after emperors died. Bohemia had two major succession wars during the seventeenth and eighteenth centuries; the winners of these wars were legitimized by the fact Bohemia had an elective monarchy.
For examples of how this form of government can be used to transition from a revolutionary republic to an Empire, consider the elections and plebiscites that elected Napoleon Bonaparte and Napoleon III in France.