Amicus curiae briefss are filed in Supreme Court cases when parties wish to bring additional information or arguments to bear on a certain case, since there is no testimony heard in front of the Supreme Court. The American Civil Liberties Union, for example, often files a brief as an amicus curiae supporting certain sides in a case.
If a case may have effects on other parties, then they may file amicus curiae briefs. For example, if a decision will affect an entire industry, even though it is brought up against only one company, other companies may file briefs as amicus curiae. Similarly, if a law in one state is under evaluation, and another state has a law that would be affected by the decision, then this other state may file a brief as an amicus curiae.
Occasionally, however, amicus curiae are not opinions on the argument or on one part of the argument, but simply an academic persepctive. For example, if the law gives deference to a history of legislation of a certain topic, a historian may choose to evaluate the claim using his expertise. An economist, statistician, or sociologist may choose to do the same.
The court has liberty to grant or deny permission of parties to file briefs as amicus curiae as it wishes. Generally, cases that are very controversial will attract a number of such briefs.