The organelles responsible for protein synthesis, and or the actual reading of the mRNA to produce the polypeptides would be the granule like organelles known as the ribosomes. Usually they can be found freely floating in the cytoplasm or on the ER.
Ribosomes are responsible for protein synthesis: Ribosomes are made up of a large subunit and a small subunit. During protein synthesis, ribosomes assemble amino acids into proteins.