What quadrilateral has two pairs of equal sides which are not opposite from each other?

The quadrilateral you are describing is a kite. A kite has two pairs of adjacent sides that are equal in length, where each pair is not opposite to each other. This property distinguishes kites from other quadrilaterals such as parallelograms and rectangles, which have opposite sides equal. Kites also typically have one pair of opposite angles that are equal.