If ABCD is a quadrilateral with a diagonal AC that divides it into two equal-area parts, then ABCD is __________

The diagonal of a parallelogram divides it into two congruent triangles.

So, 

the option 'always a rhombus' is wrong because it can be a rectangle, square or any type of parallelogram.

the option 'a parallelogram' is correct.

the options 'a kite and a trapezium' are wrong because they are not a type of parallelogram. So their diagonals don't bisect them into two parts.

Hence, the correct option is 'a parallelogram'.