Here are the quadrilaterals you are expected to know about.
Square
Rhombus
Rectangle
Parallelogram
Trapezium
Kite
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All rectangles are quadrilaterals. A quadrilateral has 4 sides; all rectangles have 4 sides; thus all rectangles are quadrilaterals.
Not all quadrilaterals are rectangles. Some quadrilaterals are rectangles, and the rest are not. However, all rectangles are quadrilaterals.
A rectangle, a rhombus, a kite, a trapezoid and a parallelogram are all examples of quadrilaterals which are not square.
Some are, and some are not. A parallelogram is a special kind of quadrilateral ... one in which every side is parallel to the one across from it. So all parallelograms are quadrilaterals, but there are also a lot of other quadrilaterals that are not parallelograms.
A quadrilateral has four sides, and so four quadrilaterals will have a total of 16 sides counted separately.