On our table, we have things that we put food on. If they are almost flat, we called them "plates." If they are like large cups that could hold soup, tazones, we call them "bowls." If they are in between, we call them "dishes."
Collectively, we call them "the dishes" (vajilla). "I have to wash the dishes." "I got them a set of dishes as a wedding present; the set has eight plates, eight bowls, eight cups, and eight saucers."
As a second meaning, we also use "dish" to mean "an item of food make by combining ingredients and cooking them," i.e. plato.