[Faceman]

Here's how it works in most democracys/republics:
A president (head of state, democratic replacement for a king) is chosen by popular/people's vote.
A prime minister or chancellor (head of government) is elected by parliament or chosen by the president. Most commonly parliament will elect (or the president will choose) the leader of the party with the most popular votes in the elections.
Note that a prime minister or chancellor is NOT the ruler/head of his country but only the leader of the government (the institution which is designed to manage the country) and in most countries he/she does not e.g. have emergency powers (e.g. is the supreme commander of the armed forces).