A shape with parallel sides but no right angles is called a parallelogram
The shape you are describing is a trapezoid. A trapezoid has two pairs of parallel sides, with one pair longer than the other. It can have two acute angles and two obtuse angles, depending on the lengths of the sides. The acute angles are typically smaller than 90 degrees, while the obtuse angles are larger than 90 degrees.
No shape does. If it has 4 sides and one pair of parallel lines it cannot have only 1 right angle, it must either have: 1) 2 right angles, 1 acute acute and 1 obtuse angle; or 2) no right angles, 2 acute angles and 2 obtuse angles; in which case it would be a trapezium (trapezoid).
It is an irregular pentagon.
A trapezoid would fit the given description providing its other two angles are acute and obtuse with all four angles adding up to 360 degrees.
pentagon
parallelogram
a trapezium
parallelogram
A trapezoid has 4 sides, 2 acute and 2 obtuse angles, and only one pair of parallel sides
a diamond does. There are two obtuse angles in it.
A parrelellegram
a diamon, pentagon, nonagon, heptagon
A parallelogram or a rhombus
this shape would be a rhombus
A trapezoid perhaps?
E.g. a parallelogram. See en.wikipedia.org.