Wikipedia
A formateur (French for someone who shapes) is a politician who is appointed by the head of state to lead the formation of a coalition government, after either a general election or the collapse of a previous government. The role of the formateur is especially important in the politics of Belgium, the Netherlands, Luxembourg, Italy, Israel, and the Czech Republic. These countries have a parliamentary system, where the executive is elected by the legislature. They also use proportional representation for elections to parliament, and have a multiparty system that makes it improbable for one party to win an outright majority. There may be several combinations of parties which might form a coalition.
The formateur most often comes from the largest party in the future coalition (although an even larger party may remain in the opposition) and generally becomes the new head of government if the formation succeeds. In a constitutional monarchy this appointment is an exercise of reserve power, and remains one of the rare moments when a mostly ceremonial monarch may play a significant political role.
In game theory and political science, the term formateur party is used to describe the party that makes a proposal in a bargaining process, most often when modeling a government formation.