The semi-presidential system is a system of government in which a president

President

A president is a leader of an organization, company, trade union, university, or country.Etymologically, a president is one who presides, who sits in leadership...

 and a prime minister

Prime minister

A prime minister is the most senior minister of cabinet in the executive branch of government in a parliamentary system. In many systems, the prime minister selects and may dismiss other members of the cabinet, and allocates posts to members within the government. In most systems, the prime...

 are both active participants in the day-to-day administration of the state

State (polity)

A state is an organized political community, living under a government. States may be sovereign and may enjoy a monopoly on the legal initiation of force and are not dependent on, or subject to any other power or state. Many states are federated states which participate in a federal union...

. It differs from a parliamentary republic

Parliamentary republic

A parliamentary republic or parliamentary constitutional republic is a type of republic which operates under a parliamentary system of government - meaning a system with no clear-cut separation of powers between the executive and legislative branches. There are a number of variations of...

 in that it has a popularly elected head of state

Head of State

A head of state is the individual that serves as the chief public representative of a monarchy, republic, federation, commonwealth or other kind of state. His or her role generally includes legitimizing the state and exercising the political powers, functions, and duties granted to the head of...

 who is more than a purely ceremonial figurehead

Figurehead

A figurehead is a carved wooden decoration found at the prow of ships largely made between the 16th and 19th century.-History:Although earlier ships had often had some form of bow ornamentation A figurehead is a carved wooden decoration found at the prow of ships largely made between the 16th and...

, and from the presidential system

Presidential system

A presidential system is a system of government where an executive branch exists and presides separately from the legislature, to which it is not responsible and which cannot, in normal circumstances, dismiss it....

 in that the cabinet

Cabinet (government)

A Cabinet is a body of high ranking government officials, typically representing the executive branch. It can also sometimes be referred to as the Council of Ministers, an Executive Council, or an Executive Committee.- Overview :...

, although named by the president, is responsible

Ministerial responsibility

Ministerial responsibility or individual ministerial responsibility is a constitutional convention in governments using the Westminster System that a cabinet minister bears the ultimate responsibility for the actions of their ministry or department...

 to the legislature

Legislature

A legislature is a kind of deliberative assembly with the power to pass, amend, and repeal laws. The law created by a legislature is called legislation or statutory law. In addition to enacting laws, legislatures usually have exclusive authority to raise or lower taxes and adopt the budget and...

, which may force the cabinet to resign through a motion of no confidence

Motion of no confidence

A motion of no confidence is a parliamentary motion whose passing would demonstrate to the head of state that the elected parliament no longer has confidence in the appointed government.-Overview:Typically, when a parliament passes a vote of no...

.

The term was first in a 1978 work by political scientist Maurice Duverger

Maurice Duverger

Maurice Duverger is a French jurist, sociologist and politician. He was born in Angoulême, Charente.Starting his career as a jurist at the University of Bordeaux, Duverger became more and more involved in political science and in 1948 founded one of the first faculties for political science in...

 to describe the French Fifth Republic

French Fifth Republic

The Fifth Republic is the fifth and current republican constitution of France, introduced on 4 October 1958. The Fifth Republic emerged from the collapse of the French Fourth Republic, replacing the prior parliamentary government with a semi-presidential system...

, which he dubbed a régime semi-présidentiel.
How the powers are divided between president and prime minister can vary greatly between countries.