A quotient can also mean just the integer part of the result of dividing two integers. For example, the quotient of 13 ÷ 5 would be 2 while the remainder would be 3. For more, see the division algorithm.
In more abstract branches of mathematics, the word quotient is often used to describe sets, spaces, or algebraic structures whose elements are the equivalence classes of some equivalence relation on another set, space, or algebraic structure. See:
Quotients also come up in certain tests, like the IQ test, which stands for intelligence quotient. In this case, your quotient is basically your score. In recent decades, as people begin to emphasize on full personal development, other similar quotients appeared. These include moral quotient, emotional quotient, adversity quotient, creativity quotient, etc.