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executive
The Senate
In the US, the US president nominates, for example, executive branch department heads and US Supreme Court justices. When his or her nominations are approved by the Senate, they take their "appointed" positions. For the most part the nominations are approved by the Senate. With that said, the president does make direct appointments. The position that heads the US Patent Office is a presidential appointment. The Senate is not involved.
Appointments to the Supreme Court are nominated by the President and approved by the United States Senate. There is a hearing, and one more than half of the Senate must approve in the affirmative for the appointment to be valid.
Checks and balances. The Executive branch has power over (checks the power of) the Judiciary branch by way of choosing who to appoint. The Legislative branch has power over (keeps in check) the Judiciary branch by way of choosing whether to confirm the appointments. The Legislative Branch also has power over (keeps in check) the Executive branch because the appointments to the Supreme Court require their approval.
The US President. Congress however has to ratify the Presidential appointments.
Executive branch
Congress can check the executive branch in a number of ways. It can override a presidential veto with a 2/3 majority. It also reviews (and subsequently confirms or denies) presidential appointments (e.g. to the Supreme Court). It also controls the national budget, and - perhaps most importantly - it has the power of impeachment, through which it can remove the President from office.
The President (Executive branch) nominates US Supreme Court justices, with the "advice and consent" of the Senate (Legislative branch), which votes whether to confirm or reject the nominee. The House of Representatives is not part of the Supreme Court appointment process.
If the US Supreme Court declares an Executive Order (Presidential action) unconstitutional, it is checking the Executive Branch.
It falls to the Senate to approve appointments made by the President of the United States. These appointments encompass the Department of Justice, ambassadors, deputy secretaries, US attorneys, and general counsels.
The Senate has the sole power to confirm Presidential appointments. The power of "advice and consent" is outlined in the Article II, Section 2, Nomination Clause.