The Supreme Court's decisions and opinions can be categorized as follows:
A Majority Opinion is a decision agreed upon by more than half of the justices. This is the ruling decision of the court, and the opinion explains the rationale behind that decision.
A concurring opinion occurs when a majority of judges have reached the same decision but arrived there with a different rationale. Multiple concurring opinions can be written to justify the same decision by the court.
A dissenting opinion is made by one of the justices whose vote was in the minority. This is an explanation of why they think the court's decision is incorrect. These opinions are not used as a way to set precedent.
Memorandum opinions are meant to make a decision in a particular case. They do not set precedent to be used in future cases. These are typically issued in cases where the law is already clear enough that no further explanation is necessary for review. Usually, they are made as a way to affirm that a lower court's decision was the correct one.
On occasion, a per curiam decision will be made. These decisions do not list the way that individual justices voted. They remain anonymous. It signifies that the decision was reached by the court, rather than by the individual justices of that court.
Originally a seriatim decision could be made. These often occurred when multiple issues are addressed by a single case. With a seriatim decision, each justice read their own opinion rather than just a single majority opinion. However, this type of opinion was discontinued in the early nineteenth century.