The Supreme Court
Our forefathers and the initiators of democracy in the United States created three branches of government in the belief they should be separate but equal with no one branch having more power than the others. So they established the Executive, the Legislative and the Judicial branches with each acting as a check and balance to the other two. Obviously the Executive Branch is the President of the United States, the Legislative Branch is composed of Congress, both the U.S. Senate and the U.S. House of Representatives, and the Judicial Branch is the Supreme Court of the United States. They are indeed separate from each other but they are not equal.
The Supreme Court decides on what the law of the land shall be based on a judicial interpretation of the Constitution and the Bill of Rights. In theory, justices should decide cases based on the law rather than politics but that's where the theory pretty much goes out the window. The reason for that is that Supreme Court Justices are appointed by Presidents and Presidents are historically political animals. Republican Presidents try their best to appoint conservative justices and Democrat Presidents attempt to appoint liberal judges. The biggest backfire of this idea came in 1953 when President Eisenhower appointed Earl Warren to the Supreme Court and the Warren Court eventually became one of the most liberal courts ever during the 1960's.
Today, because of the recent landmark decisions regarding the Affordable Care Act and the right of gays to be married in all 50 states, the Court is once again the hot-button topic across the land. Today's court is comprised of four liberal judges, four conservative judges and one swing judge, Anthony Kennedy, whose voting record is almost equally divided between liberal issues and conservative ones.
The single-most important thing a President does is appoint members to the Supreme Court because they serve for life, whereas a President only serves for four or eight years. So a President's lasting legacy is determined by the person or people he/she appointed to the Supreme Court because they will be deciding political issues, disguised as judicial ones, for the next 20, 30 or even 40 years. So, in the upcoming Presidential election, liberal voters will vote for ANY Democrat, regardless of who they are and conservative voters will vote for ANY Republican.
In the modern era, President Franklin Roosevelt appointed 8 Supreme Court Justices, Eisenhower 5, Nixon 4, Reagan 3 and Jimmy Carter none. So a President either has great influence on politics and law for generations to come as with FDR, Eisenhower and Nixon or no influence at all as with Jimmy Carter.
By the time the next President is inaugurated in 2017, sitting Justices Ginsburg, Scalia, and Kennedy will be in their 80s. Ginsburg is a liberal, Scalia is a conservative and, as already mentioned, Kennedy is often the swing vote of the court. In addition Justice Breyer will be 78.
If a Democrat appoints replacements for all of them, the court would swing to a 6-3 liberal majority. If a Republican fills all four seats, conservatives would have a 7-2 advantage.
Even if there are only two replacements, the court and the country will very likely take a sharp left or right turn and stay that way for decades to come. The only caveat here is that Supreme Court nominees must be approved by Congress, so it will be tougher for a Democrat President to get his nominees approved and extremely easy for a Republican President because both houses of Congress are currently in Republican hands.
Regardless, rest assured that a President who's a Democrat won't intentionally appoint a Justice with conservative leanings nor will a Republican President appoint a Justice with liberal leanings. Occasionally this backfires as it did with Eisenhower's appointment of Warren but a shift in one's political leanings after becoming a Justice is a rare occurrence indeed.
Appointing Supreme Court Justices is the most powerful and influential thing a President can do and the most important act of their administration.