Which quadrilateral has diagonals that always bisect its angles and also bisect each other

The one that fits your description is called parallelogram. This is a figure that is a quadrilateral with both pairs of opposite sides parallel. Have in mind that the parallelograms have opposite sides that are parallel, diagonals bisect each other, opposite sides are congruent, opposite angles are congruent and consecutive angles are supplementary. 

So what is the that you asking