Any polygon with exactly 4 sides is a quadrilateral. After all, that is what the word quadrilateral means. Quadri means 4 and lateral means side.
If no pair of sides is parallel, then the figure is a general quadrilateral or it is a kite. It is a kite if it has two pairs of adjacent sides that are of equal measure. It is a general quadrilateral otherwise.
If one pair of sides is parallel, but the other is not parallel, then the figure is a trapezoid. That is, unless you live in the UK or otherwise speak British rather than American English, in which case it is a trapezium.
If the figure has two pairs of parallel sides the figure is a parallelogram.
A parallelogram with 4 sides of equal measure is a rhombus.
A rhombus with 4 interior angles of equal measure is a square.
A general parallelogram (not a rhombus) with 4 interior angles of equal measure is a rectangle.