Recent Supreme Court rulings (decisions on the case they hear) have no been making the President happy. He really wants the Supreme Court to start making decisions he likes. Next month, one of the Supreme Court justices is retiring and the President sees this as his chance to get someone that will support his point of view.

What power is being used? ____
What branch used this power? _____
What check is in place for it? _____
What branch is able to check that power ____

The power being used is the power of appointment or nomination of Supreme Court justices.

The branch that uses this power is the Executive Branch, specifically the President.
The check in place for this power is the confirmation process, where the nominee must be approved by the Senate.
The Legislative Branch, specifically the Senate, is able to check the power of appointment by confirming or rejecting the President's nominee.