Actually, there are several different notions of order that are each useful in their own ways:
The closely related notion of a cyclic order is not actually a binary relation but a ternary relation.
The study of orders is order theory.
A mostly unrelated notion from group theory is the cardinality of the underlying set of a group, especially when this is finite. Sometimes one thinks of an infinite group as having order zero. The orders then have the natural order relation of divisibility?.
The term ‘order’ can also be used fairly generically as a synonym of ‘degree’ or ‘rank’, as in first-order logic, the order of a differential equation, etc. Of course, these various orders form a well-order, so this is not entirely unrelated either.