The answer to this question is much more complicated than one might think. In theory, after the Supreme Court rules, the executive branch carries out its ruling. This does happen quite often, but there are other things that can and do happen after Supreme Court rulings.
First of all, some rulings require only negative actions. They simply require, for example, that the government stop enforcing some law. For example, when the Supreme Court ruled in 1965 that bans on interracial marriage were unconstitutional, state governments had to stop enforcing those bans.
Second some rulings require no action because they leave the status quo in place. The recent ruling on “Obamacare” was like this. No action was required because the Court simply upheld a law that was already in place.
Third, other parts of the government sometimes fight back against Supreme Court rulings. In the case of statutory decisions, Congress can simply rewrite the law. It did this, for example, after the Court struck down federal gun free school zone laws in United States v. Lopez. It rewrote the law to get around what the Court had said. As another example, state governments tried various actions, such as abolishing public schools, to prevent implementation of school desegregation after the Brown decision in 1954.
Finally, Supreme Court decisions can lead to major political fights. Perhaps the best example of this is the Roe v. Wade decision on abortion from 1973. This case caused abortion to become a major national political issue. The fights over abortion have significantly affected the availability of abortions in the US even though the decision seemed to ensure that they would be available.
The brief answer to this question is that the executive branch enforces Supreme Court decisions. However, the truth is much more complex.