It is the culminating class of a course of study, usually taken senior year. It is generally a class where a student demonstrates all that has been learned in his or her major, usually by writing a major paper and presenting it before a academic panel (professors), doing a major project, engaging in a research project, or doing an internship of some kind. It can be a class of from 3 to up to 12 credit hours.
At some universities, a capstone class is simply the class where these projects and papers are presented to faculty and student peers. Or the capstone is a mentoring class, where the student receives some guidance from a faculty member on his or her senior project or paper. When that is the case, the class is usually only 1 or 2 credit hours. The actual work that is presented is done in other classes, and granted more credits through those classes.