How is Switzerland run?

Switzerland has a very democratic system of government.
Switzerland is a confederation of 26 semi-independent cantons. A canton is equivalent to what is called a state in English-speaking federal countries.
As many decisions as possible are made at the cantonal level. Each canton has its own constitution, parliament, laws and court system. Most of the cantons were once independent countries, and on many ways continue to act as such.

As well as regular elections, Switzerland has a form of "direct democracy" where any decision can be put to a referendum. A petition is started, signatures are collected and if enough signatures are collected there has to be a referendum which is binding. A referendum can be at the federal or cantonal level.

Switzerland has a collective head of state, the seven-member Swiss Federal Council.
The president is the presiding member of this council. Elected by the Federal Assembly for one year, the President of the Confederation chairs the meetings of the Federal Council and undertakes special representational duties. Primus inter pares, the President has no powers above the other Councillors and continues to head his or her department. Traditionally the duty rotates among the members in order of seniority and the previous year's Vice President becomes President.
The official title is "President of the Confederation" (German: Bundespräsident(in), French: Président(e) de la Confédération, Italian: Presidente della Confederazione, Romansh: President(a) da la Confederaziun).