What happens if a dissenting opinion is issued by a justice of the Supreme Court?

Answer 1

Nothing really "happens". It's published along with the majority opinion but doesn't change the majority decision in any way.

Unless the Court votes unanimously, there will generally be a dissenting opinion. A justice may even opt to write an opinion "concurring" with the judgment itself, but indicating that he has reached the same conclusion via different reasoning, or "concurring in part and dissenting in part" which means he thinks part of the majority opinion is good and part of it is not.

It's not uncommon for there to even be three or more opinons published, with various justices agreeing with each other about some parts but disagreeing on others.

For example, the majority opinion might be: the decision of the lower court is overturned for Reason 1, Reason 2, and Reason 3. A dissenting opinion may state that the decision should not be overturned at all, and an opinion concurring in part and dissenting in part may say that Reason 3 is total hogwash (they generally polite it up a bit, but that's often the general gist... reading Supreme Court opinions is sometimes more entertaining than you might think), but that Reason 1 and Reason 2 are still good, and a fourth opinion might say that Reasons 1, 2, and 3 are all wrong, but the decision should still be overturned because of Reason 4. None of this changes the basic decision itself (to overturn the lower court's ruling) in any way.