Article Two of the U.S. Constitution declares that the President "shall appoint… Judges of the supreme Court". The handling of the President's nomination is then passed on to the Senate Judiciary Committee, similar to the U.K.'s Judicial Appointments Commission, which produces a report following hearings and interviews of the prospective Justice. Lastly, a vote is held in the Senate in which the deciding verdict is confirmed: rejection or approval.
The Twenty-second Amendment means that no President can serve more than two terms, yet a Justice has no fear of reprisal as the occupation entails indefinite tenure. A large majority of Justices only stop serving upon dying. Hence, an appointment can be considered a President's lasting legacy.
Presidents largely nominate people with similar political dispositions as their own. Ronald Reagan's notable nomination of Antonin Scalia is an example of a conservative President leaving a conservative imprint upon the judiciary. Scalia is known for taking a strictly originalist view of the Constitution, has been called a homophobe by Congressman Barney Frank and is widely admired by conservatives. Due to there being no requirements for the position, technically a President can nominate anybody - arguably a great power.
The Supreme Court has a considerable impact on American life. The bicameral legislature's role is to make laws, but most of the legislation proposed in Congress is not passed. During the last Congress (the 111th), only 3% of bills were enacted. The Supreme Court's much faster process involves both the interpretation of the codified laws and the creation of case law. Its decisions have enormous impact - landmark cases, such as Brown v. Board of Education of Topeka which was central to ending racial segregation, shape the liberties and rights of Americans for decades to come. These appointments are a President's only way of influencing those outcomes.
On the other hand, Presidents can often only appoint one or two Justices. Four Presidents have appointed none at all. This contradicts the claim that these are the most important appointments a President makes - as it assumes that every President has this opportunity. Appointments can also have unintended effects, when the views of Justices evolve or the fact that there are political conservatives who are judicial liberals is not appreciated. Contrary to Reagan's nomination of Scalia, that of Anthony Kennedy largely did not fit with expectations. Despite Kennedy's conservatism, he has been at the forefront of many liberal rulings. His siding with the liberal members in Lawrence v. Texas is one example, as this led to Texas' sodomy law being struck down. Eisenhower once said that his appointment of Earl Warren as Chief Justice was "the biggest damned-fool mistake" he had ever made. The Warren Court presided over cases that advanced social progress significantly, bringing an end to state-sponsored prayer in schools (Engel v. Vitale), racial segregation (Brown v. Board of Education) and the introduction of a requirement to benefit from publicly-funded counsel to defend themselves (Gideon v. Wainwright).
There is also a limit to President’s power due to the process of interviews and the Senate vote. Harriet Miers’ nomination by Bush was withdrawn when the interviews highlighted her inexperience and it became clear that the Senate would not vote in her favour. Concerns over Miers' lack of judicial knowledge fuelled the discontent on both sides of the political spectrum. Therefore, while in theory there is no obvious criterion which prospective Supreme Court judges must meet, there are many expectations of them in practice.
Lastly, many take the view that the U.S. government's executive, rather than judiciary, is the core of power. There is no elected cabinet unlike in the U.K., hence the President's appointments of the members of his cabinet may be more important than Justices.