These terms for organizing the formal, “bricks and mortar” education procedure, have distinct, separate meanings. A curriculum (a circle of learning) is a series or set of courses built around a specialty: an English curriculum, for example, will offer courses in Renaissance drama, Romantic poetry, the history of the English novel, and the history of the English language—a group of courses preferably taken in a progressive order, leading to a thorough knowledge, even expertise, in English literature. A curriculum, then, is series of courses.
A course is a semester or quarter-long series of meetings, lectures, discussions, etc., building knowledge of one specific topic, usually circumscribed by dates and/or genre. For example, in an English curriculum, a course might be offered in Jacobean drama (1603-1625); another course might be in 19th-century fiction; another course might be in Romantic poetry. These courses, limited in time and genre, when taught in a sequence, form a curriculum, and are listed in the learning institution’s catalog.
A syllabus is a written outline of a course’s expectations—time and place of meeting (teacher and student), dates and subjects of lectures, quizzes, reading assignments (and dates to be completed), etc. Another very important function of a syllabus is that it is a “contract” between teaching institution and student, outlining grading criteria (such as percents of value of each assignment), accommodations for learning-disabled students, etc. It usually includes teacher office hours, contact information, etc.—it is the written outline of the content and sequence of learning in the course.