The quadrilateral is any 4-sided polygon. But it is the parent of a whole family of figures, each with their own distinct names and characteristics. See also Tetragon.
Quadrangle ("four angles") is another name for a quadrilateral. However, it is a word often used for an open space where people gather, say on a school or university campus. In this case the quadrangle (or "quad") is usually a rectangle or square
See also: List of Quadrilaterals.