John K. answered 09/28/20
Political Science and History Buff Ready to Help!
The Separation of Powers is obviously an important part of the U.S. Constitution. The Founders were really concerned about a branch of government having too much power, especially the Executive.
The biggest instrument the Congress as a whole has to check the executive branch is the "power of the purse" — Congress makes appropriations, creates and passes a budget, and creates taxation policy. The president/executive branch can try to influence Congress' decision-making and has veto power, but ultimately, Congress has this authority. Congress also has the sole authority to declare war, but since Vietnam, this hasn't really precluded a president from taking military action if he wanted to.
The House of Representatives has some special powers, but because Congress is a bicameral legislature, it can't do much without the Senate. The biggest tool the House has to check the President is the power of impeachment.
The Constitution grants the House the power to impeach federal officials, including the president, for "Treason, Bribery, or other high Crimes and Misdemeanors". This is an extremely vague definition that has been debated since the country was founded. Basically, if the House suspects an official broke the law, they can impeach them.
Impeachment is similar to the power a Grand Jury has in the American judicial system. It is basically an indictment against an official. The House cannot try or convict the official — that power belongs to the Senate. The House can bring up charges whenever it wants, but the Senate may or may not choose to convict.
The House also has investigatory and subpoena power. There are numerous House committees that conduct oversight of the executive branch and its federal agencies. Committee chairs, picked by the majority leadership in the House, can be really powerful and have the authority to call witnesses to testify and answer members' questions.
Sort of related, the 12th Amendment gives the House the authority to elect the president if a candidate doesn't receive a majority of electoral votes. This has happened twice — 1800 and 1824 — though the election of 1800 happened prior to ratification of the 12th Amendment.
As far as what happens if the House and Senate are controlled by two different parties? Not much gets done. Legislation can't be passed without support of both houses. The House can pass as many partisan bills as it wants, but they won't go anywhere in the Senate. The current Democratic-controlled House has passed many bills in the current session simply because it has a majority, but those bills have died in the Republican-controlled Senate.
And while the House can investigate and impeach, the Senate controlled by the other party likely won't convict.