There are two types of dominant democratic systems:
In a presidential system, the executive and legislature are separately elected and serve fixed terms. The president is both head of state and head of government, and can't be removed by the legislature (except impeachment). Prominent examples include the US, Brazil, and Indonesia.
In a parliamentary system the executive (prime minister and cabinet) are drawn from and accountable to the legislature. The government survives only while it holds the confidence of parliament and can be removed by a no-confidence vote. In these systems, the head of state is usually separate and ceremonial, being the president or monarch. Prominent examples include the UK, India, Germany.