An executive order is a specific power of the president and the executive branch as provided by the US Constitution in Article II, Section 1. This power allows the president of the United States (POTUS) the authority to create laws or determine how existing laws should be carried out. It always has to do with domestic affairs; executive agreements govern foreign affairs. Pretty much any issue in domestic affairs is fair game for such an order, except for those things that would impinge upon congressional powers, like the regulation of interstate commerce. Executive orders can be simple things like declaring a new National Holiday or a day designated to a special event, like “Take your Child to Work Day.” Since Bill Clinton’s presidency, these noncontroversial orders have been given a new name: Presidential Decision Directives.
Not all executive orders are simple or ceremonial, and some put the president in direct conflict with Congress. Some famous examples of the past include President Eisenhower’s order to enforce the desegregation of schools. Sometimes, states are — or Congress is — unwilling to enforce a law that is controversial, and under these circumstances, the president moves by executive order to see the law enforced. John F. Kennedy used these orders in a manner similar to Eisenhower, to attempt to abolish discrimination based on race for people who sought jobs, housing, or equal pay.
While the power of such an order seems broad, there are checks to it. One check is Congress's ability to overturn them, much in the same way that it can overturn a presidential veto. A two-thirds vote of both houses (the Senate and the House of Representatives) is required to overturn an executive order. This means that they can be extremely hard to overturn, since most members of Congress typically vote along party lines.
Another check to the broad power of the executive order is the Constitution. That is, the Supreme Court may review the order and weigh its constitutionality. Essentially, both the legislative and judicial branches of the government have the potential power to check or dismiss a directive, but their ability to do this may be based on the degree to which party affiliation of Congress or the courts aligns with the president.
Though use of this power has existed since the first president of the US took office, it remains a controversial one. It potentially gives the POTUS an opportunity to act in a very autocratic fashion, and checks and balances to the power through the legislative and judicial branches are only effective if these branches aren’t stacked with supporters of the president. It also gives the people of the US little recourse if they are unhappy with an order, since they have no vote or say on which orders the POTUS can pass. The only recourse given to the people is to lobby their representatives in Congress to support defeating an order and to make certain they vote for a different presidential candidate in the next election.