Elective Monarchy

"Yet I should be glad to ask how they suppose kings came at first? The question admits but of three answers, viz. either by lot, by election, or by usurpation. If the first king was taken by lot, it establishes a precedent for the next, which excludes hereditary succession. Saul was by lot, yet the succession was not hereditary, neither does it appear from that transaction there was any intention it ever should."

- Thomas Paine, Common Sense

The inverse of Hereditary Republic, this is when a monarch does not automatically inherit the throne, but is instead chosen by a group of people, usually by an elite group (typically noble families).

Film

 * Star Wars: Naboo's democratic monarchy, in which the ruler is elected and even has term limits.

Literature

 * From The Belgariad, Sendaria is this, and everyone can vote. Not even the monarch takes the monarchy seriously.
 * In A Song of Ice and Fire, royalty is normally hereditary, but a hundred years before the series takes place, the Blackfyre Rebellion occured because the king legitimized his bastard sons on his deathbed and in so doing created a massive Succession Crisis. Several years and a few thousand bodies later, the only Targaryen heirs left were either children or mentally unstable. A Great Council was formed from many of the ruling lords to choose the next king. They passed through many candidates in the Targaryen family tree before settling on Aegon V, a fourth son of a fourth son, hereafter known as Aegon "The Unlikely". After choosing the next king, the Great Council dissolved and the crown passed on through the family, though in the prelude to the War of the Five Kings, the possibility of another Great Council being formed is brought up due to the disputed heritage and validity of nearly all the contenders' claims to the throne.
 * The Ironbon, being Vikings, have something called the 'Kingsmoot' where this happens.
 * An interesting case in Mikhail Akhmanov's Envoy from the Heavens with The Empire on planet Osier, which has been stuck in Medieval Stasis for at least a thousand years, which is the reason why the protagonist is sent there in the first place - to figure out why all their efforts to secretly advance the culture have failed. The sovereign of the Empire dies, his son does not necessarily ascend to the throne. Any (male) member of the royal family may become the next ruler, provided they are popular and influencial enough within the family. In essence, the emperor is chosen by vote, but only from members of the royal house.
 * This shows that despite the name, the Empire is far from being evil. In fact, it have ruled the inhabited continent for so long (with only a few small kingdoms bordering it), that the emperors see no need to be cruel to enforce their will.

Tabletop Games

 * The Empire in Warhammer Fantasy Battle has their emperors elected by the courts of the chieftains of the provinces. Though their first emperor is now a god.

Video Games

 * Twilight Princess has the Twili, but it's never elaborated any further.
 * Ferelden in Dragon Age Origins. The king is elected by the Landsmeet, a congress of the land's most powerful nobles.
 * That's only in extreme cases, when the line of succession isn't clear. The Theirin dynasty has a long history of rule, broken only temporarily during the Orlesian occupation, and restored when Maric Theirin, the grandson of the Theirin king who was too weak to hold off the Orlesians, manages (with some help) to drive off the occupying forces and regains the crown.
 * Skyrim has a High King who is elected by the nine Jarls in the province. Before the game begins, Ulfric Stormcloak killed King Torygg and started a rebellion against the Empire with the intent of having himself proclaimed High King.
 * The Kingdom of Rhodoks in Mount & Blade games. As one of your followers notes, even though Rhodok citizens consider themselves superior to the other lands of Calradia due to having a more civilised means of government, in practise they still have a ruling elite of lords and a downtrodden peasant class, same as all the other kingdoms.

Real Life

 * The Kingdom of Poland and the Grand Duchy of Lithuania, especially after the last of the Jagiellons died without issue in 1573. It was even known as Commonwealth. With a king. One may claim it was a republic with lifelong presidential term (compare with Venice below).
 * The Holy Roman Empire (of the German Nation): the Emperor was elected by a group of four secular prince-electors (namely the King of Bohemia, the Margrave of Brandenburg, the Count Palatine of the Rhine, and the Duke of Saxony) and three archbishops (of Mainz, Trier, and Cologne).
 * Usually, anyway—the claim to the electorate of the Wittelsbach dynasty was split in the 13th century between the Count Palatine and the Duke of Bavaria, and sometimes Bavaria stepped in for the Palatinate or Bohemia (when the rival Wittelsbachs conspired with each other to exclude him on the grounds that he wasn't German). After the Thirty Years' War, the Palatinate Wittelsbachs were Protestants and the Bavarian ones Catholics, so it was decided that both would get to be electors (conveniently solving the problem of balance between Protestants and Catholics), bringing the total to eight.
 * Amusing note: In 1692 the Palatinate Wittelsbachs died out, leaving their lands and vote to their Bavarian cousins, and a ninth electorate (although there were eight electors) was created for the Duke of Brunswick-Lüneburg—who became known as the Elector of Hanover. These selfsame Electors of Hanover became The House of Hanover in Great Britain, meaning that the British monarch had a nominal hand in choosing the Emperor for about 100 years. This may have smoothed the way in getting British support for Austria (as the Austrian Habsburgs were the Emperors) in the War of the Austrian Succession, but the Austrians ditched Britain for France and Russia shortly thereafter.
 * Note that in practice, the Imperial title was hereditary, and that the Electors generally did not keep the obvious heir from the throne (although he often had to make concessions to them to make sure); instead, the system of election was used to settle disputes in the succession and select a new dynasty when the Imperial house ran out of male heirs.
 * The United Arab Emirates, in theory at any rate: although the President is supposedly elected by the rulers of the seven emirates of the UAE, it's always the Emir of Abu Dhabi who holds the position of President, and the President always appoints the Emir of Dubai Prime Minister (unless the Emir of Dubai doesn't want/can't take the job, in which case his heir apparent takes it).
 * In Denmark, the kings were selected from at least the Viking age and until 1660. With a single exception all kings (and one woman titled "Principal Lady and Husband of the North") however came from the same family, even if some spring around in the family tree were needed now and then. But it kept on, so today ruling queen Margrethe II can look back on a millenium old family tree of Danish kings.
 * In the same vein, Sweden elected it's kings until the end of the 15th century, and all free men could vote. Of course, vote for the wrong candidate and you get your teeth kicked in, but hey, it's the thought that counts.
 * The Roman Catholic Pope is indeed considered a monarch (he is the Head of State of Vatican City), and he is the last absolute monarch in Europe and one of the last in the world. He is elected by a group of cardinals.
 * Malaysia.
 * France was at some times, but each time one king sidestepped it by having his son crowned while he was still alive, and by the time the son died the monarchy was hereditary again.
 * The Most Serene Republic of Venice was one for all intents and purposes, given that the head of state (dux/doge/duke) was elected for life.
 * It is to be noted that Doges varied in power through the history of Venice. Earlier Doges were actual rulers and sometimes led fleets in battle. Later ones were essentially walking totems rather like a Japanese Emperor, a British King, or an Athenian Archon Basilus and in fact a modern British royal has far more personal freedom and may have more political influence then some Doges.
 * Venice had an elaborate system of holding several stages of elections and lotteries designed to weed out factional gamesmanship in the choice of Doge. This was necessary in vendetta prone Italy and may have contributed to Venice's success as a state.
 * The King of Cambodia is elected by a council, even if there is a successors available.
 * The Grand Master of the Order of Malta is elected for life.
 * The King of Saudi Arabia is also technically elected. Technically, because of two caveats:
 * 1) When the electors (the seniormost princes of the House of Saud) vote, the King is generally still alive, and they thus usually elect a Crown Prince, to succeed the current King when he dies. Theoretically, if the King and Crown Prince die within a very short span of time, the princes might have to elect a King, but this has never happened.
 * 2) The electors have always elected the most-senior male member of the House of Saud deemed qualified for the job (some princes are ill, uninterested, or otherwise under suspicion, and thus aren't candidates in the first place). Since the 1950s, this has meant, basically, the oldest surviving son of Abdul Aziz Ibn Saud. That's right—it's been over sixty years since the man died, and his grandsons still aren't anywhere near the throne (the Crown Prince and de facto next in line are both sons of Abdul Aziz).
 * The Roman Empire and its Byzantine successor, oddly enough, despite their reputation, made at least nominal acknowledgement of this. The Code Justinian in fact states that the Emperor was an agent of the people. In reality, the rules for Imperial succession were never quite made clear and because of that bore more then a few traces of Asskicking Equals Authority. However, the Emperor at least had to take public opinion into account. The downside was that it was the legions' public opinion that counted most, and thus military budget cuts were hard to get through. One substitute was to use soldiers for makework projects which often were useful in themselves but more importantly kept soldiers too busy sweating to be ambitious.
 * Tanists (crown princes effectively) of Scottish clans tended to be chosen by election. The custom was to prevent succession problems if the chief was killed in battle; they could not after all call a halt to a clan war just to decide who was going to be the next chief.
 * Anglo-Saxons had the Wittanagemoot, or meeting of wise men, which consisted of a meeting of the VIPs in the kingdom and was an informal Parliament. One of its powers was to debate the choosing of a new king. That power has in some sense devolved to Parliament. It is a nice legal point whether UK is a real elective monarchy, as the hereditary arrangement has been little interfered with since the first Hanovers. However, there is precedent, and at the least Parliament has proved that it can remove a monarch or even a whole dynasty should situation warrant.
 * The Norman version of the Conquest claims that both Edward and Harold had sworn the crown to William. Aside from the fact that Harold's oath was questionable, being under duress, it may well be that they had both sworn only to lobby for William and it may even be that both kept that promise. They certainly had the power to keep that promise and they didn't have the power to simply swear off the crown to a foreigner. As lobbying in a consensual forum was unknown in Normandy, that may mean that the Conquest was a matter of cultural confusion.
 * Prince Frederick Charles of Hesse was elected King of Finland by the Finnish parliament. Due to international politics concerning his bloodline he never served and Finland became a Republic instead.