A quadrilateral is a polygon with four sides.
There are many special types of quadrilateral.
A parallelogram also has the following properties:
A rectangle is a parallelogram with four right angles, so all rectangles are also parallelograms and quadrilaterals. On the other hand, not all quadrilaterals and parallelograms are rectangles.
A rectangle has all the properties of a parallelogram, plus the following:
A rhombus has all the properties of a parallelogram, plus the following:
A square can be defined as a rhombus which is also a rectangle – in other words, a parallelogram with four congruent sides and four right angles.
A trapezoid is a quadrilateral with exactly one pair of parallel sides. (There may be some confusion about this word depending on which country you're in. In India and Britain, they say trapezium; in America, trapezium usually means a quadrilateral with no parallel sides.)
An isosceles trapezoid is a trapezoid whose non-parallel sides are congruent.
Finally, a kite is a quadrilateral with exactly two pairs of adjacent congruent sides. (This definition excludes rhombi. Some textbooks say a kite has at least two pairs of adjacent congruent sides, so a rhombus is a special case of a kite.)
The following Venn Diagram shows the inclusions and intersections of the various types of quadrilaterals.