Rhombuses (rhombii) have 4 equal sides.
Both pairs of opposite sides are parallel. They all have 2 lines of symmetry (4 lines if they are a square!). All rhombuses belong to the parallelogram family.
Parallelograms have 2 pairs of parallel sides.
Some parallelograms have lines of symmetry (depending on whether they are also squares, rectangles or rhombuses), but most do not.
Trapezoids US (Trapeziums UK) have one pair of parallel sides.
Some trapezoids have a line of symmetry. Please note the differences between the definitions for US and UK.
Kites have 2 pairs of equal sides which are adjacent to each other.