"Tallerken" is what you eat your food from. It's a dinner plate.
"Plade" means 'plate' in other senses, like a metal plate or a tectonic plate. For example, a number plate (for a motor vehicle) is a "nummerplade". It can also refer to a music disc like a vinyl LP or a CD.
Examples:
"Man plejer at spiser mad fra en tallerken." (One usually eats food from a plate).
"De tektoniske plader bevæger sig i forskellige retninger." (The tectonic plates move in different directions.)
"Hun satte en plade på grammofonen" (She put a disc on the gramophone).