What is a quadrilateral that is not a parallelogram and not a trapezoid?

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.

Source: Wikipedia

A quadrilateral that is neither a parallelogram nor a trapezoid is called a kite.

A kite is a quadrilateral with two distinct pairs of adjacent sides that are congruent. In other words, a kite has two pairs of consecutive sides that have equal lengths, but the opposite sides are not necessarily equal or parallel. Additionally, kites have one pair of opposite angles that are congruent.

The diagonals of a kite are perpendicular to each other, and one diagonal bisects the other. However, unlike a parallelogram, the diagonals of a kite are not necessarily equal in length.