What is a quadrilateral that is not a parallelogram and not a trapezoid?
There are quite a few answers, here courtesy of Wikipedia.
Kite: two pairs of adjacent sides are of equal length. This implies that one diagonal divides the kite into congruent triangles, and so the angles between the two pairs of equal sides are equal in measure. It also implies that the diagonals are perpendicular.
Right kite: a kite with two opposite right angles.
Trapezium (NAm.): no sides are parallel. (In British English this would be called an irregular quadrilateral, and was once called a trapezoid.)
Tangential quadrilateral: the four sides are tangents to an inscribed circle. A convex quadrilateral is tangential if and only if opposite sides have equal sums.
Cyclic quadrilateral: the four vertices lie on a circumscribed circle. A convex quadrilateral is cyclic if and only if opposite angles sum to 180°.
Bicentric quadrilateral: it is both tangential and cyclic.
Orthodiagonal quadrilateral: the diagonals cross at right angles.
Equidiagonal quadrilateral: the diagonals are of equal length.
Ex-tangential quadrilateral: the four extensions of the sides are tangent to an excircle.