Depending on the individual shapes, you can make a large triangle, a square or any other quadrilateral, a regular or irregular pentagon or hexagon. If concave shapes are allowed, you can make any polygon upto a decagon.
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1 rhombus 4-5 triangles
Depending upon exactly where the cut is made (and the shape/type of the original trapezoid), a trapezoid cut into two pieces [of equal area] can result in: 1: two trapezoids 2: a trapezoid and a parallelogram 3: a trapezoid and a [general] quadrilateral 4: a parallelogram and a triangle 5: two triangles.
The answer depends on the relative sizes of these shapes.
trapezoid
It is not possible to have 1 parallel side since it must be parallel to another side. A four sided shape with 1 PAIR of parallel sides is a trapezium (or trapezoid).