Burma (aka Myanmar): This Asian nation has a complicated system in which three presidential candidates are chosen by the Presidential Electoral College, which consists of members of the lower and upper houses of the Pyidaungsu Hluttaw, or Assembly of the Union and members of the nation's military. The full assembly then picks a president, and the two who lose become vice-presidents. The real power in the government, though, is held by the state counselor, which is similar to a prime minister.
Burundi: This small east African nation uses an electoral system to elect representatives to its Inama Nkenguzamateka, or Senate. 36 of the body's 43 legislators are indirectly elected by a college of provincial councils that use a three-round voting system. In the first two rounds, a candidate must get a two-thirds majority, while the final round is determined by a simple majority.
Estonia: This tiny former Soviet republic on the Baltic Sea chooses its president through an even more complicated system than the U.S. Estonia's Parliament votes on presidential candidates, but if none of them manage to secure two-thirds of the vote after three rounds, an electoral college consisting of both members of Parliament and local council members chooses between the two candidates who got the highest number of votes. If that doesn't result in a clear winner, the decision goes back to the entire Parliament.
India: The south Asian nation picks its president and vice-president indirectly through an electoral college consisting of members of both houses of the Indian Parliament. Under the Indian system, the presidency is a largely ceremonial role akin to a monarch, since the office has no actual executive powers, according to this BBC article. The real authority goes to the nation's Prime Minister.
Madagascar: This island nation off the East African coast relies upon an electoral college of municipal, communal, regional and provincial leaders to pick 42 of the 63 members of its Antenimieran-Doholona, or Senate. The remainder are chosen by the nation's president.
Nepal: In the mountainous Asian nation, 56 of the 59 members of its National Assembly are elected by an electoral college of state and municipal government leaders.
Pakistan: The south Asian nation's president is selected by an electoral college, consisting of members of the Senate, National Assembly and provincial assemblies.
Trinidad and Tobago: The Caribbean island nation's president is picked by an electoral college of selected Senate and House of Representatives members.
Vanuatu: The small south Pacific archipelago's president is elected by an electoral college consisting of Parliament and presidents of the nation's six provinces.
Vatican City: The tiny European state's political leader is the Pope of the Roman Catholic Church. The Pope is elected, usually for life, by the College of Cardinals, an international group of high church leaders who were appointed by a previous Pope.