Short answers, pulled from the story.
The executive is the part of the government that executes or enforces the law and policy of a government. It can be organised as a branch of government, as in liberal democracies, or as an organ of the unified state apparatus, as in communist states.
In a presidential system, the leader of the executive is both the head of state and the head of government, and the directly elected head of government appoints the ministers. In a parliamentary system, a cabinet minister responsible to the legislature is the head of government, while the head of state is usually a largely ceremonial monarch or president.
In systems with fusion of powers, typically parliamentary systems such as the United Kingdom, the executive is often called the government and its members generally belong to the party that controls the legislature. Because the executive requires the legislature's support and approval, the two bodies are fused together rather than independent.
The top leadership roles include the head of government, often the prime minister, who oversees the administration of all affairs of state. They also include the defence minister, interior minister, foreign minister, finance minister, and justice minister, each overseeing a defined area such as the armed forces, police forces, diplomatic service, treasury, or criminal prosecutions.
In parliamentary systems, the executive is responsible to the elected legislature and must maintain its confidence, or the confidence of one part of it if bicameral. If the legislature expresses a lack of confidence, the result is either a change in the governing party or group of parties or a general election.
Yes, the executive can be the source of certain types of law or law-derived rules, such as a decree, executive order, or delegated legislation. For example, the U.S. federal legislative branch frequently delegates rulemaking, the creation of regulations, to the U.S. federal executive branch.