The three main branches/organs of government are the legislative, executive, and judiciary. Very simply, the legislative branch makes the laws, the executive enforces the laws, and the judiciary interprets the laws. Many countries today practice some form of this system of government, particularly the United States. Here, the three organs exist as a system of checks and balances to ensure that no one branch exceeds the others in power.
In the United States' legislative branch, each state is represented by two senators. The number of representatives each state receives, however, is based on population. For example, based on the latest 2010 census, California has fifty-three representatives, more than any other state in the union. Meanwhile, Vermont, Wyoming, South Dakota, North Dakota, Montana, Delaware, and Alaska each have one representative. In all, there are 435 members in the House of Representatives. There are also six non-voting House of Representative members: the resident commissioner of Washington, D.C., and the delegates of Puerto Rico, the US Virgin Islands, American Samoa, Guam, and the Northern Mariana Islands.
The six non-voting members may not vote in the full House; they can, however, vote in committees of which they are a member.
In the executive branch, the president is the head of state and the commander-in-chief of the Armed Forces. The executive branch consists of the president, vice-president, the president's Cabinet, fifteen executive departments, and hundreds of federal agencies. The president and vice-president are elected through the electoral college system. In this system, each state elects its own electors. These electors must legally vote for the presidential and vice-presidential candidate who receives the most votes in their states. The executive branch may also make treaties with other countries, but the treaties must be ratified by the Senate.
The judiciary branch is established by Article III of the Constitution. Members of the federal judiciary branch are nominated by the president and confirmed by the Senate. There are currently nine Supreme Court justices. Additionally, there are thirteen Courts of Appeals and ninety-four federal district courts. Federal justices can only be removed through a process: first, impeachment by the House of Representatives and then, conviction by the Senate. Federal justices serve for life. For more, please refer to the link provided.
There are variants of this system of government in other countries. In Ireland, for example, the Taoiseach (or Prime Minister) is the head of government. He or she is nominated by the Dáil Éireann (House of Representatives) and then appointed by the president. The president of Ireland must appoint whomever the Dail nominates.
In a country such as Saudi Arabia, the three organs (executive, legislative, and judiciary) are dominated by members of the ruling royal family. Members of the judiciary and legislature are appointed by the king. They serve at the monarch's pleasure and can be dismissed at any time.
Meanwhile, in Taiwan, there are actually five branches of government: the executive, legislative, judiciary, examination, and control. The president is the head of state and commander-in-chief of the Armed Forces. He has sole power to appoint and remove civil and military officials, arbitrate disagreements between the branches, and, surprisingly, the right to dissolve the legislative branch if the latter chooses to dismiss the premier. The premier is actually the head of the executive branch. For more, please refer to the link above.