This would be a parallelogram or a rhombus. A parallelogram is like a pushed over rectangle and a rhombus a pushed over square. All sides are equal length and parallel on a rhombus but opposite sides are equal and parallel on a parallelogram. None of the sides are joined by 90 degree angles.
The quadrilateral you are describing is a parallelogram. In a parallelogram, opposite sides are parallel and equal in length, but it does not necessarily have right angles; the angles can be acute or obtuse. Examples of such parallelograms include rhombuses and non-right-angled rectangles.
If both pairs of opposite sides are the same length, and both pairs of opposite angles are the same, the quadrilateral is a parallelogram (rhombus, rectangle, or square).
square
Parallelogram. You could also call it a rhombus if all sides are the same length, but a rhombus is technically a parallelogram.
A parallelogram.
Square
QUADRILATERAL
If both pairs of opposite sides are the same length, and both pairs of opposite angles are the same, the quadrilateral is a parallelogram (rhombus, rectangle, or square).
A Rhombus
square
A rectangle is a quadrilateral with four right angles and opposite sides that are equal in length. The diagonals of a rectangle are also equal in length and bisect each other at 90 degrees.
Parallelogram. You could also call it a rhombus if all sides are the same length, but a rhombus is technically a parallelogram.
Most quadrilaterals do not have opposite sides which are equal in length. Sadly, there is no special name for them.
A Rectangle
a pentagon
A parallelogram.
A parallelogram.